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Revista de la . '"

Unlon . "' .

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i Maternatlea Argentina

Volumen 44, Número 1 - 2003

Bahía Blanca - 2003

ISSN 0041-6932

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Revista de la Unión Matemática Argentina

CONSEJO DE REDACCION

Agnes Benedek

Instituto de Matemática Universidad Nacional del SUI�

Bahía Blanca, Argentina

Roberto Cignoli

Departamento de Matemática-FCEyN Universidad de Buenos Aires,'

Buenos Aires, Argentina

Luiz Monteiro

Instituto de Matemática Universidad Nacional del Sur,


Horacio Porta

Department 01 Mathematics University ol1/linois,

Urpana-Champaign, m., USA

Domingo Tarzia

Departamento de Matemática-FCE Universidad Austral, Rosario, Argentina.

Jorge Vargas

FAMAF Universidad Nacional de Córdoba, Córdoba,

Argentina.

Eduardo ZarantonelIo

CRICYT Mendoza, Argentina.

Luis A. Caffarelli

Department 01 Mathematics University 01 Texa at Austin,

Austin, Tx:" USA,

Carlos Kenig

Department 01 Mathematics University 01 Chicago,

Chicago, m., USA.

María Inés Plafzeck

Instituto de Matemática Universidad Nacional del SU!;


Carlos Segovia Fernández

Departamento de Maiemática-FCEyN Universidad de Buenos Aires,


Juan Tirao

FAMAF Universidad Nacional de Córdoba,

Córdoba, Argentina

Victor Yohai

Departamento de Matemática-FCEyN Universidad de Buenos Aires,


Felipe ZÓ

Instituto de Matemática Aplicada Universidad Nacional de San Luis,

San Luis, Argentina.

Volumen 44, Número 1

2003

REVISTA DE LA UNION MATEMATICA ARGENTI NA Volumen 44, Número 1, 2003, Páginas 1-16

WEIGHTED BMOcjJ SPACES AND THE HILBERT TRANSFORM

Marcela Morvidone IMAL-UNL

Güemes 3450 - 3000 Santa Fe - Argentina

Abstract

We obtain estimates for the distribution of values of functions Íll the

weighted BMOq, spaces, BMO�(R), that let us find equivalent norms. It is

also obtained that a suitable redefinition of the HilbeÍt transform is a bounded

operator from these spaces into themselves. This is achieved for a certain

cIass of weights w.

1 Introduction. A non negative function w defined on R is called a weight if it is locally integrable. We denote by 111 the Lebesgue rneasure of 1 and w(I) = J w(x)dx. The letter e

1 denotes a constant, not necessarily the sarne at each occurrence. A weight is said to

belong to the class Ap, 1 < p < 00, if there exists a constant e such that

(I�I ! W(X)d.<) (I�I! w(x¡- , ' , d.< f' oS e

for every interval 1 e R. The class Al is defined replacing the aboye inequality by

I� I J w(x) dx :::; e ess)nf w. 1

On the other hand w is said to belong to Aoo if there exist a and f3 such that O < a, f3 < 1 and for every interval 1 and every rneasurable subset E of 1, J w dx <

E f3 J w dx holds whenever IEI < alII· The staternent that w E Aoo is equivalent to

1 . w E Ap for sorne p. A proof of thcse facts may be found at [1].

Rev. Un. Mat. A¡:r;entina, Vol. 44-1

2 MARCELA MORVIDONE

A weight is said to satisfy a doubling condition if there exists a constant C such that

w(2I) � Cw(I) for every interval J e R. As it is easy to check any weight in Ac", satisfies a doubling condition . Now let us introduce the rnain function spaces which concern us in this work . Let rjJ : R+ -7 R+ be a non-decreasing function satisfying the 62 Orlicz's condition rjJ(2r ) � CrjJ(r) for sorne positive constant C and every r > O.

Definit ion 1.1 Let J be a loeally integrable funetion on R. We say that J belongs to BMO�(R) if there exists a eonstant C su eh that

w(I)�(IJI) J If(x) -f¡l dx � C (1)

I

for every finite interval 1 e R. Here f¡ denotes the average of J oveT the interval 1, i.e.,

f¡ = I�I J f(x)dl:. I

The srnallest constant C satisfying (1) will be denot�d by IIJII and it defines a nonn in BMO�(R). In case that (1) only holds for those J lying in sorne fixed interval Jo, not necessari1y of finite measure , we say that f belongs to BMO�(Io). In that case, the smallest constant C satisfying the inequality will be called Ilfll/o' We note that if rjJ == 1 the space BMOr coincides with the space BMO(w) defined by Muckei.houpt and Wheeden in [5]. We now introduce a class of weights which appears in connection with the boundedness of the Hilbert transforrn on the BMO� spaces .

Definition 1.2 Let w be a weight. We say that w E H (rjJ, (0) if thcTe exists a constant C such that

1 I1 J w(y)rjJ (Ixo -yl)d Cw(I) -- y < --rjJ (IJI) Ixo - yl2 - 1 I1 R-I

for eveTy finite intervall e R, where Xo denotes the eenter of J.

In the previous definitions we have not imposed any constraints on the growth of rjJ. It is known that if w == 1 and rjJ(t) = t{3, with j3 > 1 , the unique functions belonging to B M O� are the constant ones . In the weighted case, the spaces B M O� rnay be non-trivial for such functions rjJ: that depends on the weight W. In fact , there are examples showing this situation. However, it rnay be proved that if the function 4>S) is still increasing then BMO� is trivial for ever'y weight W.

Rev. Un. Mat. A rgentina, Vol. 44-1

WEIGHTED BMOrp SPACES AND THE HILBERT TRANSFOR M 3

Similar considerations hold for the classes H (rjJ , 00 ) . In other words, if rjJ increases

in such a way that <PS) is non-decreasing, the only weight belonging to the class is

w == O. We can also point out that if rjJ(t) grows faster than t (let us say <P¡t) is non-decreasing) but slower than t2 , then the class H( rjJ, 00 ) is non-trivial, although the weight w == 1 does not belong to it. Finally, though we will not impose additional restrictions on the functions rjJ in the statement of the theorems, it is clear that they become trivial when rjJ increases

faster than t2. We present now a pair of lemmas that will be necessary later. See [51 for a proof of the following lemma.

Lemma 1.1 Let 1 O.

Lemma 1.2 is quite similar to Lemma (4.7) in [4]' which has been adapted to this

contexto

Lemma 1.2 Let w be a weight satisfying a doubling condition. lf fE BMO� (R) then there exists a constant e such that

j' I f(y ) - J¡ldy :S ellf ll J w(y) rjJ (I xo - y l) dy I xo - y l2 I xo - y l2

R-I R-I

for every inter-val 1 e R, where Xo is the center of l .

Proof. Given an interval 1 BMO� (R), we have

l(xo , R) let lj l(xo , 2j R) . Using that f E

J I f(y) - J¡ldy = � J I J:o - y l2 � )-02j R:Slxo-yl<2J

+l R

If(y) - J¡I dy I xo - y l2

R-I

00 1 j' :S � 22jR2 I f(y) - J¡ldy

)- IJ+l -Ij

:S Cl11-1 f I;-j

I J If(y) - J¡ldy '-0 )+1 )- IJ+!

Rev. Un. Mat. Argentina, Vol. 44-1

4 MARCEL A MORVIDONE

The last inequality is a result of using the properties of 4>, the doubling condition for w and the following relations

! w(y)4> (Ixo -yl) dy = � ! w(Y)4> (Ixo -yl) dy Ixo -yl2 � Ixo -yl2 R-I - 2kR::::lxo-yl<2k+lR

> CIII-l � 2-k4> (Ihl) w (1 - 1 ) - L 1 11

k+l k

which completes the proof. O

k=O k

> Clll-l � 2-k4> (Ihl) w(1 ) - L 1 11 k, k=O k

2 Behavior of the distribution function and equivalent norms

Let f E BMO�(Io ). The question is: how does the distribution function w({x E 1 : lJ(x) -JIlw-:(x) > a} ) , for a > O and 1 e 10 , behave? An answer is given in [51 for the case 4> == 1 and it may be used in our case. The result obtained by Muckenhoupt and Wheeden is the following theorem:

Theorem. Let f be of bounded mean oseillation with weight w on 10 , that is

J If - JIj dx 5, C ! w dx; 1 1

JI = I�I ! f dx, 1

a) If w E A11 there are positive eonstants Cl and C2 su eh that


1 e 10 .

WE IGHTED BMOrp SPACE S AND THE HILBERT TRANSFO RM

for o: > O and 1 e lo. b) Jf w E Ap, 1 o:}) ::; C3 ( 1 + o:rp' w(I) for 0:'> O and 1 e lo.

5

H we denote by [1]10 = sUPw�I) J If(x) - hldx, a careful look at the proof shows lelo 1

that in the case [1ho ::; 1 the constants in the aboye estimates depend only on the Al or Ap condition for the weight w and not on f neither on the interval lo. From

this, the general case fol!ows, as usual, taking g = uL. Therefore a) and b) can be

written as a' ) Jf w E Al, there exist positive constants A and B such that

B w( {x El: If(x) - hlw-l(x) > o:}) ::; Ae -(fIlO a w(I)

for o: > O and 1 e lo. b') Jf w E Ap, 1 o:})::; D ( 1 + [j�IJ -p' w(I)

for o: > O and 1 e lo. To estimate the distribution function for f E BMO� (Io) let us take a finite interval 1 e lo, then BMO� (Io) e BMO� (l) , and the inclusion is a continuous operator, in fact, IlflII::; IlfllTo' Since by definition we have

w(I)� (III) J If(x) - hldx ::; IlflII, 1

for al! 1 e l, the fact that cjJ is an increasing function implies

[J]¡ ::; cjJ(lll) IlflII· Therefore, combining inequalities we may write

[J]I::; cjJ(lll)llflllo' for f E BMO� (Io) and for all l e lo. Final!y noting that h(x) = e-�, c > O, is an increasing function of x we may apply a' ) and use the previous inequality to obtain for f E BMO� (Io) and w E Al

_ _ 13 a _ w({x El: If(x) - J¡lw-1(x) > o:}) ::; Ae m¡;;- w(I)

B ::; Ae - <,6(11]) 1 l!II¡o a w(l)

Rev. Vil. Mat. A rgentina. Vol. 44- 1

6 MARCE LA MORVI DONE

for all a > O and 1 e lo. AIso, having that the function g(x) = (1 + �rpl with

e> O and p' > O is also increasing we may use b') for w E Ap, 1 O and 1 e lo. So we have achieved our goal of knowing the behavior of the distribution function for f E BMO� (Io). Next theorem clearly states this result and gives in addition a hint on an alternative way to prove it . Moreover, it shows that the condition on the distribution of values is not only necessary but also sufficient for an f to belong to BMO� (Io).

Theorem 2.1 Let f E BMO� (Io), i. e. , there exists a constant e such that

W(I)� (III) J If(x) - J¡ldx ::; e 1

for every interval I e lo. Then

a) Jf w E Al, there exist positive constants A and B such that

_ B '" w({x El: If - J¡ lw-l (x ) > a} ) ::; Ae tP(IIIlII/Jllo w(I)

for a > O and I e lo.

(2)

(3 )

b) Jf w E Ap , 1 O and 1 e lo. Conversely if w E Al and there exist positive constants A and e such that

(5)

for a> O and I e lo then f E BMO� (Io). On the other hand, if w E Ap, 1 a} ) ::; e' (1 + <P(�ll)) w(I) . (6)

for a> O and I e lo then f E BMO� (Io).

Rev. Un. Mat. Argentina. Vol. 44-1

WEIGHTED BMO<p S PACES AND THE HILBERT TRANSFORM 7

Proof. That the condition 1 E BMO� (Io) implies (3) and (4) has been proved by the argument given aboye . This may also be shown following the steps of the proof in [5] but changing the definition of the function A(CY, 1) there by

A(cy, 1) = w({x E 1 : 11(x) - hlw-l(x) > cyrP(lll)})· Then the same arguments can be carried out leading us to the required estirnates.

Conversely, if w E Al and 1 satisfies (5) we have

I I.f - hldx = 1 11 - hlw-lwdx I I

00

= I w({x E 1: 11 - hlw-l > CY})dCY o

00 < Aw(I) rP (111)

I e- 4>�fl) e dCY � C rP (III) o � Ow(I)rP (111) ,.

where o = �. Therefore 1 E BMO� (lo). Similarly, if w E Ap and 1 satisfies (6)

00

I 11 - h 1 dx = I w ( {x E I : 11 - h 1 W -1 > CY}) dCY J o

lOO ( ) -p' d ::; C'rP (111) w(I) 1 + rP (711) rP (I� I) o ::; O'w(I)rP (111) ,

We conclude that 1 E BMO� (Io) which completes the proof of the theorem.D

The characterization of the distribution function of the elements in BMO� given in the previous theorem allows us to introduce sorne equivalent norrns .

Theorem 2.2 Let 1 ::; p < 00 and w E Ap. Then 1 E BMO� (Io) il and only il there exists a eonstant Cr su eh that

(1 I.f - hlrwl-rdx)l/r::; CrrP (lll) (w(I))l/r (7) I

8 MARCE LA MORVIDONE

for every 1 e lo and every r su eh that 1 < r � p' and r < oo. Por every fixed r satisfying these eonditions the infimum eonstant- Cr defines an equivalent norm in BMO;(Io).

Proof. Suppose that f satisfies (7) for sorne r, 1 < r < oo. Then, by Hülder's inequality

/ 11 - f¡ldx = / 11 - f¡lw 1;" w ";1 dx I I

� (/ If - f¡lrw1-rdx)1/r(w(I))1/rl

I � Cr cjJ (111) (w(I))l/r+l/rl

= Cr cjJ (111) w(I). Therefore , 1 E BMO;(Io). Conversely, let us suppose that f E BMO�(Io). Then, if O < r < 00

/ If - f¡lrwl-rdx = / (11 - f¡lw-lrwdx I I

00 = r / ar-lw({x E l: 11(x) - f¡lw-1(x) > a} )da

o If w E Al , using Theorern 2.1 and denoting C = Ilflllo , we have

Therefore

00 / 11 - f¡lrwl-rdx � Arw(I) / ar-le-<pufllCoda I o

00 = Arw(I) / are-<p(lfllCod:

o 00

= Arw(I) / (tcjJ (III)Cre-Bt�t

o 00

= A r w(I)cjJr (111) Cr / tr-1e-Btdt o

= Arllfll�ocjJr (111) w(I).

Rev. Un. Mat. A rgentina, 1,hl. 44-1

WEIGHTEO BMOIjJ SPACES ANO THE HILBERT TRANSFORM 9

Now if w E Ap, 1 r ( 11 1) w(I),

sinee pi > r. Raising to the power 1/r both sides of the inequality we obtain (7).0

The result presented in the following eorollary will be useful latero

Corollary 2.1 Let 1 :S p < 00 and w E Ap. If f E BMO� (R) then there exists a nurnber q > 1 such that fE Lioc(R) . Moreover, for any finite intervalI we have

IlfIILq(I) :S Cllfll</>(IIl)w(I)III�-l.

Proof. Sinee w E Ap, it satisfies a reverse Holder inequality, i . e . , there exists {J > 1, depending only on p and on the Ap eonstant for w, su eh that , for every interval 1 ( ) 1/(3

I�I f ,v'dx :S I�I f w(.r.)dx,

with a eonstant e not depending on 1 (see [2]). Next let us ehoose q > 1 and s> 1 sueh that 1 < qs :S pi and �s�l

l :S {J. So the reverse Holder inequality also holds for q:�ll. Using the previous theorem, we will have

(/ IfllJdx)ljq = (/ Iflqw�w�dx)ljq 1 1

:S (/ Ifl'1Sw1-'1sdx) l/'1s (/ w (qS;l) s' dx) l/'1s'

I ¡ = (/ IflqSw1-QSdx)lj'1S(/ w'1f1 dx) s;/

1 1

Rev. Un. Mat. Argentina, Vol. 44,1

1 0 MARCELA MORVIDONE

� ellfll<p(111)(w(I))1/qs[( I�I ! w� dX):q-=-

lll� 1118;'/

1

� ellfll<p(111)(w(I))1/qSI[I �- t.- _1

_1 (W(I))1-t.-1111- qs

= ellfll<p(11I)w(I)111�-1.

for any fixed interval 1 e R, and we obtain the desired result o o

3 Hilbert Transform Let f be a measurable function in R. We define the operator

1lf(x) = lim ! [_1_ + X(y) ] f(y)dy f-¡O+ X - Y Y

Ix-yl>f

where X(y) is the characteristic function of Iyl > 1, provided the limit exists for almost every X. We denote by H f the Hilbert transform, i . e . ,

Hf(x) = lim ! -l-f(y)dy. f-¡O+ X - Y

Ix':"yl>f

lt is easy to see that if 1lf and H f both exist for almost every x then they differ by a constant . This is the case if, for example , f E V, 1 � p < oo . However 1lf may exist while H f may not o As we will see later this happens , for example, when f is a constant function. It is known that the Hilbert transform H f is bounded in L� if and only if w E Ap,l < p < 00 [3]. Moreover, for the case p = 00 it is shown in [5] that the operator 1l previously defined is bounded from L OO(w-1) = {J : Ilfw-11loo < e} in BMO(w) if and only if 4' E H(l, 00 ) n Aoo. The next theorem shows that 1lf is well defined for f E B M O� ,w E H (<p, 00 ) n Aoo and that it is also a bounded operator. In particular, if <P == 1 we have an extension of Muckenhoupt and Wheeden's result , since LOO(w-1) � BMO(w).

Theorem 3.1 Let f E BMO�(R). Jf w E Aoo n H(<p, 00 ) then there exists a constant e such that

w(I)� (IJI) ! l1lf(x) - (1lf)¡ldx ::; ell!11 for every J e R, 1 '

i.e., 1lf is a bounded operator from BMO!(R) into itself.

Rev. Un. Mat. A rgentina, Vol. 44- 1

WEIGHTED BM01/! SPACES AND THE HILBERT TRANSFORM 11

Proof. To prove that the Hilbert transform is well defined over BMO� we will show first that if e is a constant then He = o. We only need to prove this for

e = 1. In that case we have

Note that

H1 (x) = lim J f-tO Ix-yl>f

(_1_ + X(y)) dy. x-y y

lim lim J X(-R,R) (x -y) (_1_ + X(y) ) dy = o. f-tO R-too X - Y Y Ix-yl>f

In fact, if E and R are fixed numbers such that O < E < R, then

J R>lx-yl>f

1 --dy=O. x-y

AIso, if Ixl > 1, considering R > 21xl and E sufficiently small , we have

Ixl+R

J t dy = sg (x) J t dy - J t dy R>lx-yl>f R-Ixl Ix-yl<f

1Y1>1

= ± ( ln (Ixl + R) - In (R - Ixl)) - J t dy, Ix-yl<f

(8)

therefore, the first term tends to zero if R -+ 00 and the second tends to zero if E -+ O. When Ixl < 1 and R > 2 we only have the first term and taking lim we obtain the . R-too resulto On the other hand, it is easy to see that

In fact

J 1_1_ + X(y) 1 dy < oo. x-y Y Ix-yl>'

J Ix�y+X�Y)ldY= J Ix�YldY+ J l-x�-y +tldY.

Ix-yl>f Ix-yl>f Ix-yl>f Iyl<l Iyl>l

It is easy to check that the first term is finite. AIso

Rev. UI1. Mat. A rgentina, Vol. 44- 1

12 MARCELA MORVIDONE

! IX�y +tldY= ! Ix�x�IIYldY+ !

Ix-yl>- Ix-yl>f Ix-yl>f lyl>l IYI>2Ixl 2Ixl>lyl>1

lyl>l

1_1_ + �ldY x -y y

sclxl ! 1 :1 2dY+ ! (IX�YI+ltJ)dY, lyl>2lxl 3Ixl>lx-yl>f

2Ixl>lyl>1

and this shows that both terms are finite. Using Lebesgue's dominated convergen ce theorem, from (8) we obtain 1-l1 = O, as stated. In order to see that 1-lf(x) is finite almost everywhere we may write 1-lf(x) 1-l(J -f¡)(x). Let x E 1 = (-R, R) and R> 1/2 . Then

1-l (J-f¡ ) (x) = lim ! ( _1_ + x (y)) (J(y)-f¡) dy . f�O X - Y Y Ix-yl>f

= lim ! (_1_ + x (y)) (J (y) -f¡) dy + f�O . X - Y Y Ix-yl>f lyl>2R

+ lim ! (_1_ + x (y)) (J (y) -f¡) dy f�O X - Y Y . Ix-yl>f lyl<2R

= T1(x) + T2(x). For the first term note that Ix -yl > Iyl-Ixl > Iyl-R > Iyl /2 , so using Lemma 1.2 and the fact that w E H(cp, (0 ) , we have for x E 1

IT1(x)1 S l im ! 1 ( x) IIf(Y) -f¡1 dy f�O x -y y

Ix-yl>f lyl>2R

< 2R J If(y) -f¡ld - lyl2 y

lyl>2R

< 2R ! If(y) -f¡ld - lyl2 y lyl>R

S 2Rc Ilfll J w (Y��2(lyl) dy

lyl>R

S e Ilfll w(I)cp (11 1 ) 111-1 .

WEIGHTED BMOI/J SPACES AND THE HILBE RT TRANSFORM

On the other hand

T2 (x) = lim f _1_ U(y) -f¡) dy + lim f ,-+0 X -Y ,-+0

Ix-yl>' Ix-yl>' lyl<2 R 2 R>lyl>1

f(y) -f¡ d y. y

13

From Corollary 2 . 1 we know that there exists a number q > 1 such that f (y) - f¡ E Lioc' Since Hg(x) is finite for almost every x when 9 E U, considering g(y) =

X( -2 R,2 R) (y) U (y) -f¡) we condude that the first term is finite a .e . The absolute value of the second term is bounded by

f If(y)-J¡l d < f If( )-f l d < Iyl y - y T Y 00,

2 R>lyl>1 lyl<2 R because f(y) -f¡ E Ltoc' We have shown then that 1lf(x) is finite for almost every x E J. Letting R ---+ 00 our daim is completely proved. To get the norm estimate let J be any finite interval . Since 1le = O when e is constant , we have that

1lf(x) = 1lU -f¡)(x) Let g(x) = U -f¡)(x). Then 1lf(x) = 1lg(x). If 1 = 21, we put 9 _= gl + g2, with gl = gX¡ and g2 = gX(1)c where (I)C denotes the complement of J. Then

f l1lf(x) -(1lJ)¡l dx = f l1lg(x) -(1lg)¡l dx ¡ ¡

::; f l1lg1(x) -(1lg1)¡l dx + f l1lg2 (x) -(1lg2 )¡l dx ¡ ¡

= JI + J2

Since w E AXl and BMO!(R) , using Corollary 2. 1 we obtain that there exists 1] > 1 such that f E Lioc(R) and therefore gl E U(R). Then we have that 1lg1 = H gl + e and so

f l1lg1(x) -(1lg1)¡l dx = f IHg1(x) -(Hg¡)¡l dx 1 ¡

::; 2 f IHg1(x)l dx. ¡

Now, using Holder's inequality and the fact that the operator H is of strong type (q, q) for q > 1, we have

Rev. Un. Mat. Argel/tina, Vol. 44- 1


J IHg¡(x)ldx ::; (J IHg¡(xWdx)¡/qIIII/ql ¡ ¡

::; (/ IH91(xWdx)l/qIIII/ql R

::; C(J Igl(XWdx)l/qIIII/ql. R

Let us estimate IlgIIILq(dx) = (J If - f¡lqdx)l/q. ¡

¡ ¡ = 211f11Lq(T) + A2 = Al + A2•

¡

Corollary 2.1 gives us a bound for Al. Now, let us estimate A2.

A, � (¡ Ih - hl'dX) 'l.

= Clf¡ - hllJII/q

::; C I�I J If - hldx IJII/q

¡

::; C I�I J If - hldx IIII/q

¡ ::; Cw(J)e/>(IJI)IIII/q-l llfll·

We have used the fact that f E BMO�(R) in the last inequality. Therefore

where we used the doubling condition for w. Finally,

J 11lgl(x) - (llgl)¡ldx::; C Ilfll e/> (111) w(I). ¡

Rev. Un. Mat. A rgentina, \thl. 44- 1

WEIGHTED BMOI/J SPACES AND THE HILBERT TRANSFORM 15

Now, we will estimate 12 using Lemma 1.2 and the fact that w E H (cjJ, oo). Let Xo denote the center of 1 and put R = �, then we have

IHg2(x) -H.92(Z)1 = ! (_1 ___ 1_) g2(y) dy x-y z-y

Therefore

when x, z E 1 . But since

Te ! Iz-xl

:::; 1 1I I Ig2(y)l dy x-y z-y Te

:::; Iz-xl !

:::;Glll ! 1I -Irl dy

Ix -yllz -yl

11-f¡ld y Ixo -yl2

:::;Gllllllll ! w(y)cjJ (Ixo -yl) dy Ixo -yl2 Ixo-yl�R

= C!11-1cjJ(lll)w(I)11111

! IHg2(x) -(Hg2)ll dx = ! IHg2(x) -I�I ! Hg2(z) dzl dx l l I

using (9) we have

= ! II�I ! H.92(X)dz -I�I ! Hg2(z)dzl dx l I I

= ! I�II ![H92(X) -Hg2(z)] dzl dx l ¡

:::; I�I! ! IHg2(x) -Hg2(z)l dzdx,

¡ ¡

! IHg2(x) -(Hg2)¡l dx :::; 1�1111

2Glll-1cjJ (111 ) w(I)11111 ¡

(9)



that is to say

J l1lg2(x) - (1lg2)¡ldx S e ci> (IJI) w(I)lIfll· ¡

Putting together the estimates for 1lg1 and 1lg2 we obtain the desired resulto o

References [1 ] R. R. Coifrnan and C . Fefferrnan, Weighted norm inequalities fo r maximal fune

tions and singular in tegrals , Studia Math. , 51 ( 1974) , pp. 241-250.

[2] J. García-Cuerva and J. 1. Rubio de Francia, Weighted norm inequali ties and related topies, Mathematics Studies, 116, North Holland (1985) .

[3] R. A. Hunt , B . Muckenhoupt and R . L. Wheeden, Weighted norm inequali ties for the eon juga te funetion and Hilbert transform, Trans. Amer. Math . Soc . , 176 ( 1973), pp. 227-251.

[4] E. Harboure , O. Salinas and B.Viviani , Boundedness of the fraetional integral on weighted Lebesgue and Lipsehi tz spaees , Trans . Amer. Math . Soc . , 349 ( 1997) , pp. 235-255.

[5] B. Muckenhoupt and R. L. Wheeden, Weighted bounded mean oseil lation and the Hilbert transforní, Studia Math . , T . LIV. ( 1976) .

Recibido : 16 de Agosto de 2001 . Aceptado: 25 de Setiembre de 2002 .

Rev. Un. Mat. A rgentina. Ift-J/.44-1

REVISTA DE LA UNION MATEMATICA ARGENTINA Volumen 44, Número 1, 2003 , Páginas 17-32

ONE-SIDED SINGULAR INTEGRAL OPERATORS ON CALDERÓN-HARDY SPACES

S. OMBROSI AND C. SEGOVIA

ABSTRACT. In [5J we have defined and studied the 1t�:;t(w) spaces for weights w belonging to the class A;; definecl by E. Sawyer, ancl where the parameter a is a positive real number. When a is a natural number, these spaces can be identifiecl with the one-sidecl Hardy space H� (w) clefinecl in [7J. This identifieation could be used to define a eontinuous extension of a one-sided regular Calderón-Zygmund operator from 1í�:;t(w) into 1t�:;t(w), when the parameter a is a natural number. In this paper, we give a direct definition of a one-sided regular Calderón-Zygmund operator on Aa n1í�:;(w), which is valid for any real number a> 0, and we prove that these operators can be extended to bouncled operators from 1í�:;t (w) into 1t�:;t (w).

1 . NOTATION, OEFINITIONS ANO SOME PREVIOUS RESULTS

17

Let f(x) be a Lebesgue measurable function defined on IR. The one-sided HardyLittlewood maximal functions M+ f(x) and M- f(x) are defined as

1 ¡X+h M+ f(x) = sup h If(t) 1 dt

h>O x and

1 ¡X M-f(x) = sup h If(t) 1 dt.

h>O x-h

As usual, a weight w(x) is a measurable and non-negative function. If E e � is a Lebesgue measurable set, we denote its w-measure by w(E) = fE w(t)dt. A function

f(x) belongs to U (w ) , O < s ::; 00, if IlfIILs(w) = (f�oo f(x)Sw(x)dx) l/s is finite.

A weight w(x) belongs to the class A;-, 1 ::; s < 00, defined by E. Sawyer in [7], if there exists a constant e such that ( 1 lx ) ( 1 ¡X+h 1) s-l

sup -h w(t)dt -h w(t)-s::¡ dt ::; e,

h>O x-h x

for al! real number x. We observe that w(x) belongs to the class Ai if and only if M-w(x) ::; ew(x) for al! real number x. It is wel! known that if w (x) E A;(1 < s < 00), then thete exists a constant ew such that the inequality

(1)

holds for every f E U (w) (e.g., see [7] or [4]). Given w(x) E A;-, 1 ::; s < 00, we can define a number X-oo, -00 ::; X-oo ::; 00,

such that for almost every x, w(x) = O in (-00, x-oo) and O < w(x) in (x-oo, +00).

Ke:1J 'Words and phmses. one-sided weights, one-sided regular Calderón-Zygmund ker

neis, Calderón-Hardy spaces.

2000 Mathematics Subject Classification Primary: 42B20j Secondary: 42B35. This research has been partially supportecl by UBACYT 2000-2002, CONICET and FOMEC .

Rev. Un. Mat. Argentina. Vo!. 44-1

1 8 S . OMBROSI AND C. SEGOVIA

Let us fix w E A;- and let X-oo be as before. Let Lfoc(x-oo, 00) ,1 < q < 00, be the space of the real�valued functions f(x) on IR that belong locally to U for compact subsets of (x-oo, 00 ) ) . We endow Lfoc(x-oo, 00) which the topology generated for the seminorms

I f l q,J = ( 111 -1 ! I f (yW dY) l/q ,

where 1 = [a, b] is an interval contained in (x:... 00 , 00) and 111 = b - a. For f(x) in L[oc (x_?O , 00) , we define a maximal function nia (f ; x) as

n�a (f; x) = sup p-a Iflq,[x,x+pj , . p>O where a is a positive real number.

Let N a non negative integer and PN the subspace of Lfoc(X-oo, 00) formed by all the polynomials of degree at most N. We denote by EN the quotient space of

L[oc(Loo, 00) by PN· If F E EN, we define the seminorm I I F l l q,J = inf { l f l q,J : f E F} . The family of aH these seminorms induces on EN the quotient topology.

Given a real number a > O, we can write a = N + (3, where N is a non negative integer and O < (3 :s: 1. This decomposition is unique.

For F in EN, we define a maximal function Nia(F; x) as

N::a(F;x) = inf {n�a(f;x): f E F} . We say that an element F in EN belongs to the Calderón-Hardy space 1-l�:;t (w), O < p :s: 1, if the maximal function N;}:a (F; x) E lJ' (w) . The "norm" of F in 1-l�:;t (w) is defined as 1 1 F 1 1'H�:t(w) = IIN;}:a(F; x) I ILP (w) ' These spaces have been defined in [5] and, in the case that w = 1, these spaces have been studied in [3].

We say that a class A E EN is a p-atom in 1-l�:;t (w) if there exist a representative a (y) of A and a bounded interval 1 such that

i) supp(a) e 1 e (x-oo, 00) , w(I) < 00 ii) N;}:a (A , x) :s: W(I)-l/p for aH x E (Loo, 00) . In [5] it was proved the following result:

Theorem 1.1 (Descomposition into atoms ) . Let w E A;- and O 1 or (a + l/q)p > 1 if s = 1. Then, if F E 1-l�:;t (w) there exists a sequence Pi} of real numbers and a sequence {Ai} of p-atoms in 1-l�:;t (w)

. such that F = ¿ .\;Ai en EN (x-oo, 00 ) . M oreover the series ¿ '\iAi converges in 1t�:;t (w) and there exist two constants Cl and C2 not depending of F, such that

Cl 1 1F1 I��:t(w) :s: ¿ l,\dP :s: c2 1 1 F 1 1��:t(w)' As before, let a = N + (3, where O < (3 :s: 1. We denote by Aa (x-oo, 00) , the

space consisting of those classes F in EN such that if fE F then f E eN (x-oo, 00) , and there exists a constant e such that the derivative DN f satisfies the Lipschitz condition

O NE-SIDE D SINGULAR lNTEGRAL O PERATO RS 19

To sirnplify the notation, we write Aa instead Aa(x-oo, 00) . In the following lernrna we state sorne results on the rnaxirnal function N:'a (F, x) and the spaces 1í�:; (w) that we will need in this papero

Lemma 1. 2. Let F E E'J.¡. (i) Jf N:'a (F, xo) is finite for some Xo there exists a unique representative f of F

such that N:,a(F, xo) = n;'a(J, XO). (ii) F belongs to Aa if and only if there exists a constant finite C such that

N:'a(F,x)::; C for all x E (x-oo, 00) . (iii) Jf F E 1í�:; (w) and t > O, we can decompose F as F = Gt + 8t, where

N:'a(Gt,x)::; C t for all x E (x-oo, 00) and

roo N:a(8t, x)Pw(x)dx::; C { + N:a(F,x)Pw(x)dx. Jx_oc J {xE(x_oc,oo):Nq,,,,(F,x»t}

Prooj. Part (i) is Lernrna 2 . 2 in [5] , part (ii) is Lernrna 3.10 in [5] and part (iii) is Lernrna 4.3 in [5]. O Corollary 1.3. The set 1í�:;(w) n Aa is dense in 1í�:;(w).

We say that a function k in Lfoc(lR - {O }) is a regular Calderón-Zygrnund kernel, if there exists a finite constant C such that the following properties are satisfied:

(a) l !g<IXI<M k(x)dxl ::; C holds for aH é and M, O < é < M, and there exists

lirng--+o+ !g<lxl<l k(x )dx. (b) I k ( x ) I ::;

I�I' for all x # O .

(c) Ik(x - y) - k(x)1 ::; C ly l lx l -2 for aH x and y with Ixl > 2 1y l > O . We observe that (b) irnplies that for r > O,

(2 ) 1 . Ik(y)ldy::;Cl ly l-1dy::;C'. r::;lyl::;2r r::;lyl::;2r

A regular Calderón-Zygrnund kernel with support in ( - 00 , O) will be called a one-sided regular Calderón-Zygrnund kernel. In [ 1] H. Airnar, L . Forzani and F. Martín-Reyes proved that the class of these kernels is not ernpty, in fact , the kernel

(3) sin(1og Ix l ) k(x) = Ixl log Ix l X(-oo,O) (x),

satisfies the conditions (a) , (b) and (c) . We denote

K f (x) = V.p. J k(� - y)f (y)dy = lirn 1 k(x - y)f(y)dy, g--+O+ Ix-yl>g

the singular integral operator associated with k(y), and by K* f (x) the rnaxirnal singular integral operator given by

(4) K* f (x) = sup 11 k(x - y)f(Y)dy l · g>O !x-y!>g

The foHowing result can be found in [ 1].

Rev. Un. Mat. Argentina. \lb l. 44-1

20 S . OMBROSI AND C. SEGOVIA

Theorem 1.4 ( [l}). Let w E A;-, 1 < s < 00, and let k be a one-sided regular Calderón-Zygmund kernel. Then,there exists a finite constant C such that

J IK* f (xW w(x)dx ::; C J If (xW w(x)dx holds for all fE U(w).

Let n be a non negative integer , we will say that k(x) is a regular kernel of order

n, if k E cn away the origin, and

(5) I Di k (x)1 ::; IX��I' for every i = 1 , 2, . . . , n and every x 1= °

Lemma 1.5. The kernel k (x) defined in (3) is regular of order n, for every n 2: O.

Proof. We denote g(t) = si�t and f(t) = log It l . For x < 0, we get

k (x) = - (g o f(x) ) Df(x). Now, since Df(x) =�, we have that

Dif (x) = (_l )i-l (i -1)! [Df(x)Ji , for every natural number i. Arguing by induction it is eásy to see that if n is natural number, then Dnk(x) is given by a sum of n + 1 terms of the way

Ch,nDhg o f(x) [Df(x)¡n+1 , where Ch,n is a constant and ° ::; h ::; n. Then, since Dhg (t) E VJO for every non negative integer h, the lemma follows . O

2. DEFINITION OF ONE-SIDED REGULAR CALDERÓN-ZYGMUND OPERATORS ON

THE CLASSES 1í�:� (w) n Aa

We will assume in the sequel that w E A;-, where (a + l/q)p 2: s > 1 or (a + l/q)p> 1 if s = 1; and without loss of generality, we will assume that the number X-oo associated to the weight w is less than zero.

Lemma 2.1. Let a = N + 1 and let F E 1í�:�(w) n Aa. Jf fE F then

(6) IDN+1f(x)1 ::; Nia(FiX) for every x E (x_oo,oo) The proof of this lemma is similar to the proM of Theorem 4 in [2] , and it will

not be given here .

Lemma 2.2. Let F in Aa and Xl E (x-oo, 00 ) . Jf f (y) is the representative of F such that N:'a (F, X l ) = n-;'a (j, X¡), then

(7)

I Dif(y)1 ::; C I INia (F ; · ) 1100 Iy - xt!a-i holds, for i = 0, 1, . . . , N and y E (x_oo,oo).

Rev. Un. Mat. Argentina, Vol. 44- 1

ONE-SIDED SINGULAR INTEGRAL OPERATORS 21

Proof. The proof of this result is a corollary of the proof of Lemma 4.2 in [5]. In fact , with the notation of that lemma, if we consider t = IINi,,(F; .) 1100' then F coincides with the class G that appear there. Then (7) follows from estimate (24) of Lemma 4.2 in [5]. O

Let us fix a function cP ECo, ° ::; cp(y) ::; 1, supp(cp) e [-2 , 2] and such that cp(y) == 1 in [-1 , 1]. Let r > 0, and Xl E IR. We denote

(8) ( y - Xl ) CPx¡,r(y) = cP -r -.

Then, the support of CPx¡,r(y) is contained in [Xl - 2r, Xl + 2r] and cp(y) = 1 in [Xl - r, Xl + r]. Moreover, we have that

(9)

for every non negative integer i. If Xl = 0, we denote CPo,r(y) by CPr(Y)'

Lemma 2.3. Let a = N + 1, and F E 1i�:�(w) n A". Let f(y) be the representative of F such that n�" (f, O) = Ni" (F, O). Jf k(y) is a one-sided regular CalderónZygmund kernel, then

.lim 11 k( -Y)Dif(Y)DN+l-iCPj(Y)dy l = 0, for i = 0, 1, .. . , N, J-++OO

and CPj(Y) = cp(j), where cP is the function that was fixed before . Proof. By Lemma 2 . 2 , it follows that Dh f(O) = O, for h = 0 , 1 , ... , N. Then, by the Taylor's formula and Lemma 2 . 1 , we obtain

::; C 101 N:',,(F; ty)dt lyIN+l-i.

Prom the last estimate, since supp(k) e (-00, O) and supp(DN+l-icpj) e {j ::; Iyl ::; 2j}, we have that

(10) 11 k( -y)Dif(y)DN+1-iCPj(y)dy l ::; C l2j

IDN+l-icpj(y)llk(_y)llyIN+l-i 101 N:',,(F;ty)dt dy. Rev. Un. Mat. A rgentina, Vol. 44-1

By (9) , we have that I DN+l-i<j>j (y) IlyIN+1-i::; G, if Iyl ::; 2j. Prom this fact and by (10), we obtain

1I k( _y)Di!(y)DN+1-i<j>j(y)dyl ::; G 11 l2j Ik( -y)1 Nio:(F; ty)dydt

1fj 2j 1 2j =G 1 llk(-y)INio:(F;tY)dYdt+G lfjl,k(-y),Nio:(F;tY)dYdt

= Sl(j) + S2(j) By (2) , it follows that the inner integrals in S¡(j) and S2(j) are bounded by

I I Nio:(F; ·)1100 l2j Ik(-y)J dy ::; GJ,

and therefore SI(j ) --+ O, when j --+ +00. As for S 2(j ) , vve will see that

[l2j Jk(-y)J Nq,o:(F;tY)dY] --+ O.

Using condition (b) of k, changing variables and by H6lder's inequality, if SI > S 2:: 1 and for t > 1/j, we get

(11) [2j Ik( -y)J Nio:(F; ty)dy::; [ . Izl-l Nio:(F; z)dz Jj Jtj<z<2tj

. � Since w-;} E A�, by the version for M- of (1), we have that JZ>I.w�z�!'l(z)dz::; Gw, then since

( [ Nio:(F; Z ) SlW(Z )dz)l/s ::; I I Nio:(F; . ) I I� ( [ Nio:(F; z ) Pw(z )dZ )I/s , Jtj<z<2tj . ' Jtj<z<2tj tends to zero for each t � O, we obtain S 2(j ) --+ O, when j --+ +00. O

Lemma 2.4. Let F E 1t�:�(w) nAo:,and let f(y) be a representative of F. Let k be a one-sided regular Calderón-Zygmund kernel of order [a] + 1. Jf we define

(12) gj(x) = . N . - .

v.p. J k (x - Y)f(y)<P�(y)dy - t; J Dik(-y)f(y)(<j>j (y) - <PI (y))dY:; ,

where <Pj(Y) and <PI (y) are given as in (8), then there existslimj-+oo 9j in Lioc (x-oo, 00). Proof. If we denote fo the representative of F such that nt,o: (Jo,.o) = N;J:o: (F, O) , we have that f (y) = fo (y) + P (y), where P (y) is a polynomial of degree at most

ONE-STDED SINGULAR INTEGRAL O P ERATORS 23

N. Let us fix an interval 1 = [a, b] e (x-oo , 00) , and we consider a natural number l such that 1 e [- l/2 , l/2] . Then, for every x E 1 , and if j > l we have that

N . ( 13) gj (x} - g¡ (x) =

j[k (X - y) - L j

Dik (-y) �; ]f (y) (<pj (y) - <p¡ (y) )dy , ,=0

We will prove that the limit of the right hand si de of ( 13) exists . We consider two cases , the first when a is not a natural number, Le . , a = N + f3 where O < f3 < 1, and the second when a = N + 1 . In the first case , if x E 1 e [-l/2 , l/2] ' since supp ( l - <p¡ ) e I y l 2:: l , by Taylor 's formula, (5), Lemma 2 . 2 and the estimate I P(y) 1 � G( ly l + l )N , we get the following estimate for the right hand side of ( 13)

(14) 1 k(x - y) - t Dj k( -y) �!i I f (y) 1 (1 - <p¡ (y) ) dy

Iy l >¡ i=O

1 xN+l

� I DN+l k (�x - y) l l f (y) 1 dy (N + 1) 1 Iy l >¡ .

� G¡ l I �x - y l - (N+2) I fo (y) + P (y) 1 dy Iy l> l

� G¡ I I N;a (F; . ) 1100 1 l y l - (N+2) ly IN+,6 dy + G¡ l l y l - (N+2) ( Iy l + l )N dy < oo . I y l> ¡ I y l>¡

Therefore, by Bounded Convergence Theorem the right hand side of (13) converges to

when j --t oo. We observe that in this case , i .e . , when O < f3 < 1, it is enough to assume that F E Aa to prove the lemma.

In the second case, i . e . a = N + 1, in order to show that the limit of the right hand si de of (13) exists , we have to consider the cases f (y) = P (y) and f (y) = fo (y) . For the case f (y) = P (y) we argue as before. As for the case f = fo , we can write the right hand side of ( 13) as

(15)

j N+l Xi [k (x - y) - � Djk( -y) i! ] fo (y) (<pj (Y ) - <p¡ (y) )dy

j xN+l + DN+lk( -y)fo (y) (<pj (y) - <p¡ (y) )dy (N + 1) ! '

For the first term of (15) , proceeding in the same way that for f3 < 1, we see that this term converges to j N+l Xi [k(x - y) - L Djk(-y) i! ] fo (y) ( l - <p¡ (y) )dy .

;=0 Integrating by parts, we obtain that the second term of ( 15) coincides with

( _ l )N+l j k ( _y)DN+l [Jo (y) (<pj (y) - <p¡ (y ) ) ] dy .

Rev. Un. Mal. A rgentina. \ir)I. 44- 1

24 S. O MB ROSI AND C. SEGOVIA

By Leibnitz 's formula, and since supp (k) e ( - 00 , O) , the integral aboye is equal to

N+1 21 ( 16) ¿ CN,i 1 k( _y)Difo (y)DN+l-i (<pj (Y) - <pl (y) )dy

i=O 1 N+1

+ ¿ CN'i ! k (-y)Difo (y)DN+l-i<pj (y)dy . i=O y>21

If J > 2l , the first sum in ( 16) is equal to

( 1 7)

N 21 ¿ CN,i 1 k ( -y)Difo (y)D(N+l -il <Pl (y)dy i=O 1 121 + 1 k ( _y)DN+l fo (y) ( l - <Pl (y) )dy .

By (2) and Lemma 2 . 1 , the last term is bounded by

C I I DN+l fo l l oo 1211 k ( -y) 1 dy ::; C I I DN+1 fo l l oo ::; C I INia (F ; . ) 1 1 00 .

On the other hand, taking into account Lemma 2 .2, the inequality I D (N+l -il<pl (y) 1 ::; Cl- (N+ l-il and (2 ) , we obtain that each term of the sum in (17) is bounded by

121 1 k ( -y) I I Difo (y) I I D(N+1-il <pl (y) 1 dy ::;

C I I Nia (F ; . ) 1 1 00 j21 1 k ( _y) l l y I N+l-i Z- (N+l-il dy ::; C I I Nia (F ; . ) 1 1 00 .

As for the second sum in ( 16) , by Lemma 2 .3, the terms corresponding to i < N + 1 converge to zero , and the term JY>21 k( _y)DN+l fo (y )<pj (y) dy converges to Jy>21 k ( _y)DN+l fo (y) dy , in fact the pointwise convergen ce of the integrand is clear , and by Lemma 2 . 1 , for 8 1 > 8 2: 1, we have that

1 I k ( -y )DN+1 fo (y)<pj (y) 1 dy ::; 1 IY I - I IDN+1 fo (y ) 1 dy l y l >21 y>21 ::; ( r Nia(F;y)SI W(Y)dy) I/S1 (1 IYI-s� W-* (Y)dy) l/s� J1yl >21 y>21

!1.::E (1 ) 1/ SI ::; Cw,l I INia(F; · ) I I�I Nia (F; y)Pw(y)dy < 00 ly l >21

Then, lim 9j (X ) exists in Lioc (x-oo , (0 ) . J �OO o

Taking into account the notation of the previous lemma, for F E 1t�:; (w) n Aa , if f(y) is a representative of F and k is a one-sided regular Calderón-Zygmund kernel of order [a] + 1 , we define

( 18) Kof(x) = lim 9j (X) J �OO

� ,ti.');, [v ,p, J k (x - Y)f(y)�, (y )dy - t, J D;k (-y)f (y) (�, (y) - �l (Y) )d<; l ' Rev. Vil. Mal. Argemina, Vol. 44- 1

ONE-SIDED S INGULAR INTEGRAL OPERATORS 25

where the limit is taking in the sense of Lioc (x - oo , 00 ) . In Lemma 2 . 4 we have proved that for x E 1 = [a , b] e [- l/2 , l/2] , ( 19 ) Kof(x) = lim gj (x) ' J --->OO

� g/ (x ) + J [k (X - y) - t J Dik(-y):: j f(y) ( l - �/ (y) ) dy ,

where g¡ (x) = V.p. J k (x - y)f (Y)eP¡ (y)dy - L�o J Dik ( -y) f(y) (eP¡ (Y) - ePl (y) )dy * . Lemma 2.5. Let P(y) a polynomial of degree at most N , and let k (y) be a regular Calderón-Zygmund kernel of order N + 1 , then KoP(x) coincides with a polynomial of degree at most N in (x - oo , 00) . Proof. Without loss of generality, we can assume that P(y) = yn where O :S n :s N. Let us fix a natural number l , and let x E [- l/2 , l/2] n (x-oo , 00 ) . Then, from ( 19) , we have that

KoP(x) � v.p. J k (x - y)yn�/ (y)dy + J [k(X - y) - t Dik ( -y) :: 1 yn ( 1 - �/ (y) )dy

( 20) N J Xi + L Dik ( -y) yn (cp¡ (y) - ePl (y) ) dYi! = Sl (X) + S2 (X) + S3 (X) , i=O

where S3 (X) is a polynomial of degree at most N. Since k (y) is a regular CalderónZygmund kernel of order N + 1 and yncp¡ (y) E eff, it easy to see that

( 2 1 ) DN+1S1 (X) = J k(x - y)DN+1 [yncp(y) ]dy . As for S2 (x) , we can derive under the integral sign, in fact for h = O, 1 , 2 , . . . , N + 1 ,

and I y l > l , by Taylor 's formula and (5) , we obtain that

N i D� [k (x - y) - L Dik(-y) �! ] :s e I DN+1 k (�x - Y) l l x IN+1 -h :s e¡ ly l -N-2 ,

i=O then

< el r l y l n-N-2 dy < oo . J1y l>¡

Therefore DN+1S2 (x) = J (D�+l [k (x - y)]) yn (l - cp¡ (y) ) dy and integrating by parts, we obtain

DN+1 S2 (X) = J (D�+l [k (x - y) ] ) yn ( l _ cp¡ (y) )dy

= J k (x - y) D:+l [yn ( l - eP¡ (y) ) ]dy = - J k(x - y) D:+1 [yncp¡ (y) ] dy , Then, from (20) , and since S3 (x) is a polynomial of degree at most N, we have that DN+l (KoP) == O , and the conclusion of lemma follows . O

Rev. Un. Mat. A ¡gelltina, v()l. 44 -1

The previous two lernrnas enable us to give the following definition:

Definition 2. 6 . Let k be a one-sided regular Calderón-Zygmund kernel of order [a] + 1 . Let F E Aa and iJ, in addition, a is a natural number we assume that F also belongs to '}-{�;� (w ) . Then, we define K F the class in E'fv of the function

(22) Kof(x) =

;Ii.� [v .P. J k(x - y)f (y),p; (y) dy - t J V'k( -y)f(y) (,p; (y) - 'P. (Y) )d<: ] , where f (y) is a representative of F.

This definition rnakes sense , since by Lernrna 2 .4 we have that for each representative of F, the lirnit in (22) exists in the sense of Lroc (x-oo , 00 ) and by Lernrna 2 . 5 the class K F do es not depend of the representative f of F. Furtherrnore, if Xo E (Loo , 00) and if we define

(23 ) Kxof (x) =

�irn [v .p. J k(x - y)f (Y)cPxo ,j (y)dy J -+OO

�J i (x - XO ) i - i:o' D k (xo - y)f (y) (cPxo,J (y) - cPxo , l (y) )dy i! ] ,

where f is a representat ive of F. Routine cornputations show that Kxof (x) differs from K of (x) in a polynornial of degree at most N, and therefore K F is also the class of Kxof (x) . For x E [a , b] e [xo - l , xo + l] , arguing as before in order to obtain ( 19) , it follows that

(24) Kxof(x) =

i (x - xo )' [ N ' ] gxo , I (X) + J k (x - y) - f; 1 D k(xo - y) i! f (y) ( 1 - cPxol (y) )dy ,

where

9xO ,I (X) == V .p. 1 k(x - y)f(Y)cPxo ,l (y)dy -N ) ' "1 i (x - Xo ' � D k (xo - y)f (y) (cPxo , l (y) - cPxo , l (y) )dy i! . i=O

3 . MAIN RESULTS

Theorem 3.1. Let w E A; and O 1 or (a + l/q) p > 1 if s = 1 . Let K be the operator a¡sociated with a one-sided regular Calderón-Zygmund kernel k(x) of order [a] + 1 9�ven in the Definition 2. 6. Then, K can be extended to a bounded operator from '}-{�:� (w) into '}-{�:� ( w) .

If a is not a natural riurnber, Theorem 3 . 1 is a consequence of Corollary 1 . 3 and of the following result :

Rev, Un. Mat. A rgentina. Vol. 44- 1

Theorem 3 . 2. Let F E Aa , where o: = N + (3 is nat a natural number, i. e . , O < (3 < 1. Let K be the operator associated with a ane-sided regular Calderón-Zygmund kernel k (x) of arder N + 1 given in the Definitian 2. 6. Then

where C is a finite constant not depending on F.

Proof. Let us fix X l E (x-oo , 00) and p > O. Let f(y) be the representative of F such that N:a (F; Xl) = n�a (J, Xl) ' Then, for X E [X l , X l + ¡J , from (24) and associating conveniently we have that

N .

Kx¡ (J (1 - CPX ¡ ,p ) (X) = ¿ J Dik(Xl - y)f (Y)CPx¡ , l (y) dy (X �!X l ) �

i=O N .

(25) - ¿ J Dik(Xl - y)f (Y)CPx¡ , l (Y)CPx¡ ,p (y)dy (x �!

X ¡ ) '

;=0

+ J [k(X - y) -t D'k(Xl - y) (x � ,Xl) ' l (1 - �" ,, (y ) )f (y)dy

= Q(xl , x) - A + B .

The integrals in Q (Xl , X) are finite. In fact , by Lemma ( 2 . 2 ) and since supp(k) e ( -00, O) , we obtain

Then, Q(Xl , x) is a polynomial of degree at most N. By (5) and taking into account that suPP (k (X l - Y)CPX¡ ,p (Y) ) e [Xl , Xl + 2p] , we obtain that each term in A is bounded by

Rev. Un. Mat. A ,xentilla, Vol. 44- /

28 S. O MBRO SI ANO C. SEGOVIA

As for B, by Taylor 's formula, (5) and sinee /3 < 1 , we obtain that it is bounded by I

J . ( t+l

DN+l k (X I + B(x -'- Xl ) - y) ( l - cPX¡ ,p (y) )f (y)dy X(�: 1 ) !

Them, froÍn (25 ) , (26) and (27) , we obtain that for X E [X l , Xl + '¡ ] , (28) I Kx¡ U( l - cPX¡ ,p) (x) - Q (XI , x ) 1 ::; CN::cr (F; XI )Pcr .

Now, taking into aceount that cPX ¡ ,P has a bounded support and eonsidering (23) , we have that

(29) Kx¡ UcPX¡ ,p) (x) =

V .p. J k(x - y)f (Y)cPX ¡ ,p (y ) )dy N .

+ L J Dik (X I - y)f (Y)cPX ¡ ,p (y) ( l - cPX ¡ , I (y) )dY (X � !xd .

i=O Arguing as in estimate (26) , we obtain that the sum in (29) is bounded by CN::cr (F;'XI )pcr . As for the first term, sine e supp (k ) e ( - 00 , O) and taking into account that the operator K is bounded in Lq , we obtain

¡�¡+PI4 Iv .p. J k (x - y)X(X¡ ,oo) (y) f (y)cPx ¡ ,p (y) )dy l q dx

< C J I X(x ¡ ,oo) (x) cPX¡ ,P (x) f (xW dx ::; C ¡� ¡ +2P I f (xW dx ::; CN::cr (F; x ¡ ) qpcrq+l .

Thus

(30)

Therefore, from (28) and (30) we obtain that ¡X ¡ +PI4 . IKxJ(x) - Q (XI , xW dx ::; CN::cr (F; XI ) qpcrq+ l ,

X ¡ whieh implies the eonclusion of the theorem. o

We observe that Theorem 3 . 2 gives a proof of the classie result that singular integral operators associated with regular kernels map Acr into Acr .

As we have already mentioned if a is not a natural number , then Theorem 3 . 1 is a eonsequenee of Theorem 3 . 2 . If a is a natural number we eould prove Theorem

3 . 1 from the identification between 7-{�:;t (w) and the one-sided Hardy spaces H� (w) (see [5]) . However , we give here a direct proof, which follows from Theorem 1 . 1 and the following lemma:

Lemma 3 .3. Let w E A;- and O 1 or (a + l/q) P > 1 if s = 1 . Let a = N + 1, and let K be the opemtor associated with a one-sided regular Calderón-Zygmund kernel k (x) of order N + 2 given in the Definition 2. 6. Then, if A is a p-atom in 1-{�:t (w) , we have that

(3 1 )

where e is a finite constant no t depending on A.

Proof. Let a(y) be the represe�tative of A with compact support , such that supp (a) e 1 , where Nia (A ; x) :::; w (I) - l/p . Without 10ss of generality we can suppose that 1 = [O , rj . We will prove the following estimates: let Xl E (x-oo , 00) then

(i) If X l � [-2r , r] , N:a (KA; Xl ) :::; e (M+X¡ (X l )t+l/q w (I) - l /p , ,

and ( i i ) If Xl E [-2r , r],

N:a (KA ; X¡ ) :::; e [w(I) - l/p + I K* (DN+ la) (X l ) l ] ,

. where K* is given in (4) . . Let us consider ( i ) . The function Ka(x) = lime:->o+ l¡y-x l>e k(x - y)a(y)dy i s a

representative of KA. Since supp(k) e (-00, 0) , if Xl > r, we have that Ka(x) = O for X 2: X l , this implies ( i ) for Xl > r. Now, we assume that Xl < ' - 2r . We will argue as in the proof of Theorem 3 .2 . Let us fix p > O , and we as sume that X E [X l , X l + ¡] , then

K(a( l - cPX ¡ ,p )) (X) = J k (x - y)a(y) ( l - cPX¡ ,p (y) )dy .

By Tay10r 's formula, we have that

( 32 ) K(a( l - cPX¡ ,p) (x) N . N .

" J . (X - X ¡ ) ' "J . (X - Xl ) ' = L. D'k(X l - y)a(y)dy i r - L. D'k (Xl - y)a(Y)cPx ¡Ay)dy, i r i=O i=O

+ J DN+ l k (X l + B(x - Xl ) - y) ( l _ cPX ¡ ,p (y))a(y) dy (X

(� :)�:l In the same way as in the proof of Theorem 3 . 2 , we can see that the first sum in the right hand side of (32 ) is a polynomial of degree at most N, that we denote Q(Xl , x) .

We observe that since nia (a, -r) = Nia (A, -r) :::; w (I) - l /p , we have that

(33)

Rej!. Un. Mat. A ¡;r;entil1u. Vol. 44- /

Let us suppose first that p 2: IX 1�-r , and therefore that p 2: 1:f 2: � . Then, by the eondition (5) , sinee supp (a (y ) ) e [O , r] and (33) , we obtain that

(34) Jo (x - xd ¡r C o D'k (X l - y)a (y)<px l ,p (y)dy o , :S I I i+ l l a (y ) 1 dyp' z . o Xl - Y pi ¡r r",+l r",+l

:S C� l a (y) 1 dy :S �w(Irl/ppi :S aH w (I) - l/pp"' . I X l l O I Xl l I X l l Arguing in a similar way, we get

(35) X - Xl J ( ) N+l

[DN+l k (X l + B (x - x¡ ) - y)] ( 1 - <pxl ,p (y) )a(y)dy (N + 1 ) !

:S C lr I X l - y l -N-2 [1 - <PX1 , P (y)] l a(Y) 1 dypNH :S C C:l l ) "'H w (I) - l/pp"' .

Now, if p < I X 1�-r , sinee X l < -2r we have that [X l - 2p, X l + 2p] n [O , r] = 0 , This implies that

( 36) J Dik(Xl - y)a(y)<pxl ,p (y)dy = O.

On the other hand, sinee p < I X 1�-r and X E [X l , Xl + ¡ ] , for any y E [O , r] , we have

p I X l l I Xl + B(x - x¡ ) - y l 2: I Xl l - I x - xI I - r 2: I Xl l - 4 - r 2: 4 ' Then, arguing as before , we get

(37) J [DN+l k (X l + B(x - x¡ ) - y)] (1 - ePxl ,p (y ) )a(y)dy (X(� �G�l

1 r ( r ) "'+1 :S C I X l I N+2 Jo l a (y) l dypN+l :s C I Xl l w (I) - l/pl-" .

Thus, from the estimates (34) , (35) , (36) and (37) and sine e 1:1 1 < 1 , we obtain

1 ¡X l+P/4 ( r ) aq+l (38) -1 I K(a ( l - ePXl p) (x) - Q(Xl , xW dx :S C -1 -1 w (Itq/p P",q+ XI ' X l If p < IX I!-r , the supports of a (y) and ePX¡ ,P (y) are disjoint and therefore K(aePxl 'p ) (x) = O. If p 2: I X I�-r 2: I X41 1 , sinee K is bounded on Lq and by (33) , we get

1 ¡X I + ¡ q C ¡r q raqHw(I) -q/p (39) aq+l I K(aePx l ,p ) (x) 1 dx :S I I ",q+l l a (x) 1 dx :S C

I I ",q+ l . p X l Xl o Xl Then, from (38) and ( 39) , we obtain

1 ¡X I + .e -1-/ ( 4 I K a (x) - Q(Xl , xW dX) l/q :S C [M+X¡ (X l ll",H/q w (I ) - l/P , pa+ q X I

which implies ( i ) . Now, we prove ( i i ) . Let Xl E [-2r, rJ . Let f (y) E A, sueh that nt,,,, (f ; x ¡ ) =

Nq:", (A; Xl ) ' Let p > O and X E [Xl , Xl + ¡] . In the proof of Theorem 3 . 2 we

Rev. Un. Mat. A rgentina. Vol. 44- 1

ONE-SIDED SINGULAR INTEGRAL OPERATORS 3 1

saw that Q(Xl , x ) = 2:;:'0 J Dik (Xl - Y)f (y)tPx¡ , I (y)dy (x��¡ ) ' is a polynomial and furthermore

Kx¡ (f ( 1 - tPX¡ ,p) (x) - Q(Xl , x)

�! i (x - xd = - � D k (X l - y) f (y)cjJX ¡ , l (y) tPX ¡ ,p (y) dy i ! i=O

N i (40) + ! [k (X - y) � L Dik(Xl - y) (x � !X l ) ] ( 1 - tPX ¡ ,p (y) ) f (y ) dy

i=O = 11 + 12 •

Proceeding a s i n estimate (26 ) we obtain that 1 11 1 i s bounded by Cw (Itl/p pa . ( )N+ ¡

Subtracting and adding DN+ l k (Xl - y) x(��I) ! in the integrand of h and arguing as in estimate (27) , we get that

(4 1 ) 1 12 1 :::; Cw(I) - I/ppa + I ! [DN+1k (Xl - y)] (1 - tPX¡ ,p (y ) ) f (y )dy l pO .

Integrating by parts and by Leibnitz's formula, we have

I ! [DN+l k(X l - y)] ( 1 - tPX ¡ ,p (y) ) f (Y)dY I = I ! k (X l - y)DN+1 [ ( 1 - tPX¡ ,p (y ) ) f (y) ] dy l

N+ l (42) < L CN , i ! k (X l - y)DN+1-if (y )Di ( 1 - tPX¡ ,p (y) ) dy

i=1 + I ! k (X l - y)DN+l a(y) ( 1 - tPX¡ ,p (y) ) dy l ·

For i � 1 , the support of D i ( 1 - tPX¡ ,p(y) ) is contained in {y : p :::; Iy - xI I :::; 2p} . Then, since supp (k ) e (-00, O) , I DitPx¡ ,p (y) 1 :::; CiP-i and by Lemma 2 . 2 we get that the sum in the second line of (42 ) is bounded by CW(I) -l/p . As for the last summand of (42 ) , using Lemma 2 . 1 , we obtain that it is bounded by

I r I k (x l - Y) I J DN+l a (y) J dy l + 1 1 k (X l - y)DN+ 1 a (y)dy l Jp<ly-x ¡ 19P ly-x¡ I >2p :::; Cw (Itl/p r I k (x I - y) 1 + sup 1 1 k (X I - y)DN+la(y)dy l Jp< ly-x ¡ I :5:. 2p p>O ly-x¡ I >2p

(43) :::; Cw(Itl/p + K* (DN+la) (x l ) ' Then, from (40 ) , (4 1 ) and (43 ) , we obtain for x E [X l , X l + ¡ ] that

IKq (f ( 1 - tPX¡ ,p ) (x) - Q(Xl , x) 1 :::; C [w (I) - l/p + K* (DN+ l a) (x ¡ ) ] pa . On the other hand, proceeding as in the proof of (30) in Theorem 3 . 2 , we get

¡X¡+P/4 I Kx¡ (ftPX ¡ ,p (xW dx :::; Cw (Itq/ppOq+ 1 ,

X ¡ Rev. Un. Mm. A rgentina. Vr)I. 44- 1

32 S. OMBROSI AND C . SEGOVIA

Thus, we have that

(lXI+P/4 ) l /q X I

I KxJ(x) - Q (X l , xW dx :s C [w (I) - l/p + K* (DN+1 a) (x l ) ] pa+1/q ,

which implies ( i i) . Finally, we will see that ( i ) and ( i i) imply the lemma. By ( i) and (1) , we obtain

1 N;a (KA; x)Pw (x)dx :S Cw(It1 J [M+XI (X l ) ] (a+l/q)p w (x)dx :S C. (x-oc ,oo)nx�[-2r,r] By ( ii) , Holder 's inequality and Theorem 1 . 4 , we get that

1 N;a (KA; x)Pw (x)dx :s Cw + r K* (DN+1a) (X 1 )Pw(x)dx (X_oc ,oo)nxE[-2r,r] J -2r

< Cw + (J K* (DN+1 a) (xdP(N+2)w (X)dX) N�2 (J:2r

W (X)dX) 1- N�

2

I ( r P (N+2) ) N+2 I < Cw + Cw Jo (DN+1a) (X 1 ) ) w (x)dx w ( [-2r, r] ) l - N+2 :s Cw,

which concludes the proof.

REFERENCES

o

[ 1 ] H. Aimar , L. Forzani and F. Martfn-Reyes, On weighted inequalities lor singular integrals. Proc. Amer . Math. Soco Volumen 125 , number 7 , 1997, 2057-2064.

[2] A. P. Calderón, Estimates lor singular integral operators in terms 01 maximal lunctions. Studia Math. 44 (1972 ) , 563-582.

[3] A. B. Gatto, J . G . Jiménez and C . Segovia, On the solution ol the equation I:!..mp = I lor I E HP. Conference on Harmonic Analysis in honor of Antoni Zygmund, Volumen I1, Wadsworth international mathematics series, 1983.

[4] F. J . Martfn-Reyes, New prools 01 weighted inequalíties lor the one-sided Hardy-Littlewood maxímal lunctions, Proc. Amer. Math. Soco 1 17 (1993) , 691-698.

[5] Sheldy Ombrosi, On spaces associated wíth primitíves 01 distributions in one-sided Hardy spaces. To appear in Revista de la Unión Metemática Argentina.

[6] L. de Rosa and C . Segovia, Weighted HP spaces lor one sided maximal lunctions, Contemporary Math. , volumen 189 , ( 1995) 161-183 .

[7] E. Sawyer, Weighted inequalities lor the one-sided Hardy-Littlewood maximal lunctions, Trans. Amer. Math. Soco 297 ( 1986 ) , 53-61 .

S. Ombrosi .

Depto. de Matemática, Universidad Nacional del Sur , Bahia Blanca, Buenos Aires , Argentina.

e-maiZ: sombrosi@criba. edu. ar

C. Segovia. Depto. de Matemática, FCEyN, Univ. de Buenos Aires, Ciudad Universitaria

( 1428) , Buenos Aires , and IAM, CONICET, Argentina.

e-mail: segovia@iamba. edu. ar Recibido : 8 de Marzo de 2002. Aceptado : 1 de Octubre de 2002.

Rev. Vil. Mat. A rgentina, Vol. 44- 1

REVISTA DE LA

UNJON MATEMATICA ARGENTINA Volumen 44, Número 1 , 2003, Páginas 33-4 1

LINEAR COMBINATION OF A NEW SEQUEN CE OF LINEAR POSITIVE OPERA TORS

P.N. AgrawaI"

and Ali J. Mohammad**

33

ABSTRACT. In the present paper, we study the appro x i mat ion of unbounded eont inuous funet ions of exponential growth by the l i near eombi n at ion of a new sequenee of l i near pos i t i ve operators . First , we d i seu ss a Voronoskaj a type asymptot i e fo rm u l a and then obtai n an error est imate i n terms of the h igher order modu lus of e o n t i n u i ty o f the funet ion be i n g approx imated.

1 . INTRODUCTION In [ 1 ) we i n trodueed a new sequenee of l i near pos i t i ve operators M il to appro x i mate a e l ass of unbounded eont inuous funet ions of exponent ia l growth on the i n terval [O , co ) as fo l lows : [et a > O and ! E Ca [O, 00 ) = ( f E C[O, 00 ) : I! ( t) I :::; M e a t for some M > O } . Then,

� ( 1 . 1 ) Mil (f (t ) ; x ) = nI pll,v Cx) fqll , v- ¡ ( t ) ! (t )dt + ( 1 + x) -

n! CO) ,

v= 1 O (n + V - I ) v -n-V e - Il l ( nt ) v

where pIl V Cx ) = x ( I + x) , x E [O , oo ) , an d qn . v ( t ) = , I E [O, co ) . , v V i

T h e spaee Ca [O, oo) i s normed by 1 1! l le = sup 1 ! (t ) l e -a l ,f E Ca [O, oo ) . A l te rn at i ve l y , a O::;I <�

the operator ( l . l ) may be wri tten as Mn (f (t ) ; x) = fWn Ct , x)! ( t )dt , where the kerne l O

KEY WORDS : Linear pos i t ive operators , Linear eombi nat ion , Modu lus of eont inu i ty , S imul taneous approx imation.

Rev. Vn. Mat. A rgentina, Vol. 44- 1

34 P. N. AGRAWAL AND ALI J. MOHAMMAD

00 Wn ( t , x) = n ¿ P n ,v (x) q n ,v- I ( t ) + ( l + x) -n

o(t ) , o(t ) being the Dirac-delta funct ion. v=1

The operator ( 1 . 1 ) was studied for degree of appróx i mation in s i m u l taneous approximation i n [ 1 ] . It turned out that the order of approx imation of the operator ( 1 . 1 )

is , at best, O(n - 1 ) howsoever smooth the function may be. Therefore, in order to improve the rate of convergence of the óperators ( 1 . 1 ) , we appl y the techn ique of l inear combinat ion in troduced by May [4] and Rathore [5] to these operators . The approx imation process is defined as :

Fol lowing Agrawal and Thamer [2] , the l i near combinat ion M il ( J , k , x) of M d ll ( J ; x) , I j = 0 , 1 , . . . , k is defined as :

( 1 .2 ) 1 M n ( J , k , x) = -L1

M don (J ; X)

M dlll (J ; x) . . . . . . . . . . . . . . . .

M d n ( J ; x) . k

dO - 1 do -2 do -k d - 1 I d -2 I d -k I

dk -1 dk -2 d k r-k where do , d I " ' " d k are k + 1 arbi trary but fixed d ist inct posi t ive integers and L1 is the

Vandermonde determinant obtained by repl ac ing the operator co lumn of the abo ve determinan t with the entries 1 . On s impl ification, ( 1 .2 ) is reduced to

k ( \ . 3 ) M n ( J , k , x) = ¿ C( j, k ) M djn ( J ; x) ,

j=O

rr ] , k # O where C(j , k ) = i=O d j - di

¡ k d .

1 '" ] 1 , k = 0

The obj ect of the present paper is to show that by ták ing (k + 1 ) /h l i near combinat ion of

the operators ( 1 . 1 ) , O(n - ( k + I ) ) rate of convergence can be achieved for (2k + 2) t imes

continuous ly d i fferentiable functions on [0, 00) . AIso, the determinant form ( 1 .2 ) of the

l i near combi nat ion makes the determination of the polynomials Q ( 2k + I , k , x) and

Q(2k + 2, k, x) occurrin g i n the fol lowing Theorem 1 of th is paper qui te easy .

2. DEGREE OF APPROXIMATION Throughout our work, let NO denote the set of nonnegat ive integers,

O < a l < a 2 < b2 < bl < 00 and 11 . lba,bl ' the sup-norm on C[a, b] . To make the paper

self contained, w,e restáte below two lemmas from our paper [ 1 ] .

LINEAR COM B INATION OF A NEW S EQUENCE

Lernrna 1 . Let the mIl! order moment (m E NO ) for the operators ( 1 . 1 ) be defi ned by �

T/l ,m (x) = M/l ( ( t - x)m

; x) = n ¿ P/l ,V (x) fq/l , V- l ( t )( t - x)m dt + (_x)m ( 1 + x) -/l •

Then T/l ,O (x ) = 1 , T,l , l (x ) = O and

v= l o

nT/l ,m+ l (x ) = x( l + x)T�,m (x) + mT/l,m (x) + mx(x + 2)T/l ,m_l (x) , m � l . Further, we have the fol lowing consequences of T/l ,m (x) :

( i) Tn ,/11 (x) i s a polynomi aI i n x of degree m, m 1= l ; ( 1' 1' ) f [O ) T ( ) - O( -[ Im+l ) 1 2 ] ) • or every x E , oc , Il ,m x - n ,

35

( i i i ) the coeffic ients of n-k i n T/l , 2k (x) and T/l , 2k-l (X ) are (2k - I ) ! ' {x(x + 2 )Y and

C xk (x + 2) k - l (x 2 + 3x + 3) respecti vely, where C i s a constant depend ing on l y on k a n d ! ! denotes the semi-factorial function.

Lernrna 2. Let 6 and r be any two pos i t i ve real numbers and [ a , b ] e (0 , 00) . Then, for any m > O we have,

sup n¿ p/l,v (X) fqn , v- l ( t ) e Yl dt = O(n -m ) . xE [a ,b ] v=l I I-xl;::"

First, we pro ve the Voronoskaj a type asymptotic resul t for the operator M n ( f , k , x) .

THEOREM 1 . Let f E Ca [O, oc) and f i2k+2 ) exis ts at a poin t X E [O, oc) . Then

(2 . 1 )

and (2 .2)

. k + l [ ] 2�2 f lm) (x) hm n M n (f , k , x) - f(x) = L.. Q ( m , k , x ) /l�� m=k +2 m '

l i m n k+ l [ M n (f , k + I , x) - f (x) J = O ,

where Q(m, k , x) are certain polynomials i n x of degree m . Moreover, k

Q(2k + l , k , x) = (�1) C ;/ ( x + 2) k- l (x 2 + 3 x + 3 )

and

TI d) )=0

(_ l) k { }k + l Q(2k + 2, k , x) = -k-- (2k + l) l ! x (x + 2) ,

TI d ) )=0

where C is a constant dependi n g on ly on k . Further, i f f I 2k + l ) ex i sts and i s abso lute ly cont inuous over [O , b] and f I 2k +2 ) E L� [O , b l ,

then for any [ e , d ] e (O , b) there ho lds

Rev. Un. Mat. A I:�el1tin{{, Vol. 44- 1

(2.3 ) I I M n (J , k , x) - f (x)11 5, M n -(k+ I ) [ l lf l l + !!f ( 2k+ '2.l ! ! ] , C[c.d] Ca L,. [O.b j

where M is a constant independent of f and n . Proof: S ince f < 2k +2 l ex i sts at X E [0, 00 ) , i t fo l lows that

2 k+2 j' ( m l ( ) f - " x m 2 k+ 2 ( t ) - L.. I ( t - x) + E( t , X) ( t - x) , m=O m .

where E( t . X ) � O as t � x . In v iew of M il ( \ , k , x) = 1 , we can wri te

2k+2 f (m l ( ) n k+ I [M ,, ( J , k , x) - f ( x) ] = n k+ 1 I x M n ( ( t - x) m , k , x) . m=1 m !

k k+1 " C ' 2k+2 + n ¿ ( j , k ) M djn (E(t , X)( t - x) ; x ) )=0

= 1 I + 1 '2. , say. Using Lemma 1 , we h ave

T PI ( x) P2 ( x) p[m l 2 ] ( x) djn .m ( x) = (d , n ) [ (m+l l / 2] +

(d , n ) [ (m+l l l 2 ]+ 1 + . . . + (d n) m-I '

J J J for certa in po l ynomia ls P¡ , i = 1 , 2 , . . . , [m / 2] in x of degree C learl y,

k I C( j , k ) Td} I1 , 11I (x) )=0

PI (x) P2 (x) l[m I 2] (x) 1---'-'---,,::- + + . . . + -'---'--(don ) [ (m+ l l / 2] (don) [ (m+l l I 2 J+ I (don) m-: I

do

at most m .

- 1 do -2

PI (x) P2 ( x) l[m l 2 j ( .t) ----'--- + + ' " + -=----"--= t, ( d 1 n ) [ ( I1l+ l l l 2 ] (d l n ) [ (m+ l l / 2 ]+1 (d l n ) I1l- 1 d -1 1 d l -

2

dk

( 2 .4)

= n -( k + l l ¡ Q( m, k , x) + o( l ) } , m = k + 2 , k + 3 , ' : ' , 2k + 2 . 2k+2 f (m ) (x)

So, 1 1 is determined by I Q(m, k , x) + o( l ) . m=k+2 m !

-1 d k -2

1 -k (. O

d -k I

The express ion for Q ( 2k + 1 , k , x) and Q(2k + 2, k , x) can be eas i l y obtai ned from Lemma 1 in (2 .4) . Hence in order to prove (2. 1 ) it suffices to show that

Rev, Un. Mat. A rgentina. Vol, 44- 1

LINEAR COMBINATION OF A NEW SEQUENCE 37

1 2 � O as n � oo . For a given e > O , there ex i sts a 6 > O such that I € ( t . x)1 < E ,

whenever It - xl < O , and for I r - xl � o , there ex is ts a constant K > ° such that Ic( t , x) 1 ( t - x) 2k +2 � K e IX1 •

Let cp r5 ( t ) be the character ist ic function of the interval (x - 6, x + 6) , then k

1 / 2 1 � n k +1 I ICu, k) 1 M djn ( IE( t , x) 1 ( t - x) 2k+2 cp 15 ( t ) ; x) j=O .

k + n k+ 1 I I C(j, k ) 1 M d jn ( Ic(t , x) 1 (t - x) 2k+2 ( 1 - .p r5 ( t » ; x) := 1 3 + 1 4 .

j=O

Again, us i ng Lemma 1 we get 1 3 � E n k+I (± IC ( j , k ) ll max {Tdjl1 . 2k +2 (X) }< K I E . j=O 05.J5.k

Now, app l y i n g Schwarz i nequal i ty for integration and then for summation and Lemma 2 we are led to

k 1 4 � K n k + 1 I l c( j , k ) I M dJ I1 ( e U ¡ ( l - CP 15 (t » ; x ) = n k + I O ( n -m ) , for any m > O .

} =O = O(n k + l -m ) = 0( 1 ) for m > k + l .

S i nce E > ° i s arbi trary, i t fo l lows that 1 3 � O for suffic ien t ly l arge n . Combi n i ng the est imates of 1 3 and 14 we conclude that 1 2 � ° as n � 00 • The assert ion (2 .2) can be

pro ved I n a s imi l ar manner as Mn ((t - x) m , k + l , x) = 0(n -(k+2 ) , for al l m = k + 3, k + 4, . . . ,2k + 2 . Now, we sha l l prove (2.3 ) . Let 't' (t) be the characteri s t ic funct ion of [O,b ] , then M 1 1 ( (f , k , x) = M 11 ( 't' ( t ) (f (t) - f (x» , k , x) + M 1 1 ( ( 1 - 't' ( t » (f (t) - f ( x » , k , x)

:= 1 5 + 16 ' Proceedi n g as i n the estimate of 14 , we have for al l .1 E [e , d ] ,

1 6 � I lf ll e O(n -m ) , where m > O . a

From the h ypothes i s on f , we can wr i te , for a l ] t E [O , h ] and X E [e , d ] ,

2k + 1 f ( i ) ( t) . l I f(t ) - f (x) = I . ' ( t - x) ' + f( t - w) 2k+ 1 f ( 2 k+2 ) ( w) dw .

. ;= 1 i ! ( 2k + 1 ) ! x

Therefo re 2k + 1 f ( i ) (x ) i 15 = 2: . , M n ('t' (t ) (t - x) , k , x) i= 1 l .

I

+ 1 M n ('t'(t) f( t - w) 2k+ 1 f ( 2k +;2) ( w) dw, k , x) (2k + 1 ) ! x

Rev. Un. Mat. A rgentina, \t{)1. 44-1

38 P. N. AGRAWAL AND AL! 1 . MOHAMMAD

2k +1 f U) (x) { i i } = ¿ . , M n ( (t - x) , k , x) + M n ( (\f'(t ) - I) ( t - x) , k , x) i= 1 l .

t + 1 M (!P(t) f( t - w) 2k+ ' f ( 2k+2 ) ( w) dw k x ) (2k + 1 ) !

Il ' , x

2k + 1 f ( i ) (x) := ¿ . , {/ 7 + I s } + 19 '

i= 1 l .

In v iew of (2.4) , we h ave 17 = O(n- ( k+ I » , un i formly for al l x E [c , d ] . S i nce tp ( t ) i s the character ist ic funct ion of [O, b ] and X E [c , d ] , we can choose a > O such th at I t - xl ::: a . Using Lemma 1 , we have Is = O (n -(k+ I » . Again , app ly ing Lemma 1 , we get

1 1 /9 11 C[ a .h ] ::; K 2 n -( k+ l ) I lf ( 2k+2) 11 L.., [O,h l ' Combin ing the estimates of 1 7 - 19 , we have

1 1 /5 1 1 ::; K3 n - (k+ , ) ( 2f ' l lf ( i ) 1 1 + I lf ( 2 k+2) 11 l . i=1 C[ a ,h ] L.,, [O.h ]

Now, app ly ing Goldberg and Meir [3 ] property, the requi red resu l t i s immedi ate. In our next theorem we estimate the degree of approximation of M n U , k , x ) to f ( x ) i n terms o f the h i gher order modu lus o f continu i ty of f . Theorem 2. Let f E Ca [O, oo) . Then, for suffic ien t ly l arge n , there e x i sts a c o n stant M i ndependent of n and f such that

(2 .5 ) 1 1 f l l < M ) . - 1 / 2 b -( k + l l l l 11 \ M n U , k , . ) - C[az ,hz ] - L 0)2k+2 ( j , n , a l , I ) + n f c" j . 2k+' �

Proof: For f E Ca [O, oo) , the Steklov mean fl],2k+2 (X) E C - of ( 2k + 2 ) order l S defined as

1l -( 2k +2 ) 1l /f2 1l /

f2 r k -( 2k+2

) (2k + 2 J

1 2nk +2 fTj , 2k +2 ( x) = ( J . , . (- 1 ) f.. 2k + 2 f (x) + f ( x) duv , 2k + 2 -Tj / 2 -Tj / 2 ¿ uv k + 1 v = 1

k + 1 v = 1

where (k + 1 ) 2 r¡ < min{az - al , bl - bz } and ¿j�r i s the r t}¡ symmetr ic d ifference ope rator

defi ned by : 2k+2 (2k + 2J 2k+2

f.. �I( Zk+2 ) f (x) = � (_ I ) i i f (x + ( 2k + 2 - i ) �uv ) .

Then the func t ion fl] , 2 k + 2 (x ) has the fo l lowing propert i es : (2 .6 ) 1 1 Pk+"' l l l -pk+2) f b fl] ,2k+2 ::;M l r¡ - úJ2k+2 ( , r¡, al ' 1 ) ; C[a2 ,h2 ] .

Rev. Vil. Mat. A rgelltilla, Vol. 44- 1

LINEAR COMBINATION OF A NEW SEQUENCE

(2 .7 ) I lf - fr¡, 2k+2 11c¡a2 ,b2 1 '.5, M'2 úJ'2k+2 (J , !J, a\ , b\ ) ; (2.8 ) I lf11 , 2k+2 11 c¡a2 ,b2 1 '.5, M 3 1Ifll e[al ,� 1 '.5, M 4 1 1fll e" '

39

where M 4 = M 3 e � , Mi ' S are certain constants depending on k on ly and

úJ2k+2 ( J , 7], a\ , b \ ) i s the modu lus of continuity of order 2k + 2 correspond i n g to f :

úJ2k+2 (J , !J, a \ , b\ ) = sup 1.0. �Jk +2 f (x) l . [ /¡ [:>/] x,x+( 2k+2) /¡E [al '� 1 Now, in order to pro ve (2.6), notice that

k (2k + 2J 2k+' (- 1 ) k + 1 7] - fn , 2k+2 ( x)

11/2 11 / 2 [ 2k+2 J (2k + 2J 2k+2 k ( 2k + 2 J 1 2k + 2 = -) 2 -) 2 � (- 1 ) i

f (x + (k + l - i ) � U v ) + (- l ) k + l } ( x ) !]dll v

= I _1 [ i��2 (- 1)

i (2k : �J f (X + Ck + l - i)2��v )] 2!j2dUv i;tk+ \

r¡ 1 2

= f -/] 1 2

11 / 2 = f -11 1 2

S ince

/] / 2 [ k (2k + 2J 2k+2 -)

2 � (_ 1 ) 1 i f (x + (k + 1 - i) � U v )

2k+2 . (2k + 2J 2k+2 ] 2k+2 +

i=t.

2C- 1 ) 1 ¡ f (x + (k + l - i) � Uv ) D duv

11 1 2 k . (2k + 2J { 2k+2 -)

2 � (- 1 ) 1

i f (x + (k + l - i ) � Uv )

d 2k+2 11 / 2 11 / 2 [ 2k+2 2k+2 ] 2+2 2k +2 f f f(x + ¡: >v ) + f (x - ¡: >v ) IT duv = 2 .0. �( 2k + 2 ) f ( x ) ,

d x -11 / 2 -11 1 2 v=\ v=\ v= \ and úJ2k + 2 ( J ; lk + 1 - ¡ 1 7]) '.5, Ik + 1 - il úJ2k +2 (J ; 7]) , we have,

1 1 ( 2k+2 ) 1 1 T] -( 2k +2) k i (2k + 2J -( 2k+2 ) f 11 , 2 k+2 e[a b 1 = (2k + 2J I (- 1 ) ¡ 2.0. ( k + \-i ) f ( x)

2 , 2 1=0 C¡a l ,hl 1 k + 1

Rev. Un. Mat. A rgentilla, VO!. 44- 1

Tl-( 2k +2) k (2k + 2) � (?k ? ) 2I . (k + l - i) 0J2k + 2 ( f , Tl, a ¡ , b¡ )

- + - i =O 1 k + I

and thus (2 .6 ) fo l l ows. From the defi n i t ion of f r¡,2k + 2 ' we have

,, / 2 ,, / 2 2k+2 I f - f" , 2k+2 1 � ( 2k: 2) f . . . f � ��:�+2) f (x) I1 duy -,, / 2 -,, / 2 ¿ llv y=1

k + 1 v = !

� M I OJ2k+2 ( f ; Tl (k + 1 ) , a ¡ , b¡ ) � (k + 1 ) M ' OJ2k +2 ( f ; Tl, a l . ' b ¡ ) =M 2 OJ2k+2 (f ; Tl, a ¡ , b¡ ) for al l x E [a 2 , b2 ] ,

which proves (2.7). The proof of the inequal i ty (2 .8) is triv ia l and therefore we omit i t . Now, we shal l prove (2.5). we can write M n (f , k , x) - f (x) = M n (f - fr¡,2k+2 ' k , x) + ( fr¡, 2k+2 (x) - f (x»

+ (M n (fr¡,'f.k+2 , k , x) - fr¡.2k+2 ( X» = I ¡ ( x) + 1 2 (x ) + I J (x ) , sayo From (2 .7) we have

1 1 / 2 1 I c [a� .b2 1 � M 2 OJ2k+2 (f ; 1], a l , b ¡ ) = M 2OJ2k + 2 (f ; n - 1 / 2 , a l ' h¡ ) .

Next, proceed ing as i n the est imate of 14 i n the prev ious theorem, we h ave

and 00

k • I / ¡ (x) 1 � I IC(j , q fWd}n (t , x) lf (t ) - f",2k+2 ( t )I d t

j=O O

fWd ¡n ( t , x) lf ( t ) - fr¡,2k+2 ( t ) l d t = f + f O ! r-x!�o ! r-x!>o

� llf - fr¡ ,2k +2 1 1 + Km n -m I lf l l c , for al l m > o , C[aro,bro] a where, 0 < min { a 2 - a ¡ , b¡ - b2 } . Hence, again in view of (2 .7)

1 1 / ¡ I I C[a2 ,b2 1 � M 2 OJ2k +2 (f ; n - 1 /2

, a l ' b¡ ) + K m n -m l li l l ca • Final l y, in order to est imate I J (x) , we observe that by Taylor expans ion

1 k +2 f ( i ) ( ) . -I r¡, 2k+2 x i i ( 2 k+2 ) 2k + 2 (2.9) ir¡ ' k + 2 ( t ) = . ( t - x) + fr¡ 2k + ' (�) (t - x) , , -i=O l ! ( 2k + 2) ! ' -

where � l ies between t and x . Operating M ( . , k , x) on (2 .9) and separat i n g the in tegral in to two parts as i n the est imation of I¡ (x) , from Lemma 1 and (2 .4) we are led to

LINEAR COMBINATION OF A NEW SEQUENCE 4 1

2k+2 < lA' -(k+ l ) � I lf U ) 1 1 K -1M I lf 11 - 1Y1 5 n L.. 1l , 2k+ 2 + 1M n 1l . 2 k + 2 •

i= 1 C[" , .b , ] Cu Using [ 3 ] , we get

I lf�:�k +2 1I C [G2 ,b2 1 � M 6 ( 1Ifr¡, 2k +2 II C[ a" b2 ] + 1 !J�,22�:�La, .h, ] ) ' and choos i ng m � k + 1 , we have further that

11M 1 1 (fr¡, 2k +2 , k , . ) - fr¡, 2k +2 11 C[ Q2 .b2 ] � M 7 n -( k+ l ) (1Ifr¡, 2k +2 1I ca + I lf �.�kk:� II C[G2 .b2 ] ) ' Now, app l y i n g (2 .6) , (2.8 ) and the defin i t ion of fr¡, 2k+2 we get :

\ \ 1 3 [ [ C [G2 ,b1 ] � M 8 (W2k+2 ( f ; n -1 / 2 , a l ' b1 ) + n -( k + l ) [ [f [ l ea ) . Combini ng the est imates of 1 1 ( x ) - 1 3 (x) w e obtai n (2 .5 ) .

ACKNOWLEDGEMENT The authors are ex treme 1 y gratefu l to the referee for mak in g usefu l suggest ions l eadi ng to a better presentat ion of the i r papero

References : [ 1 ] Agrawal , P.N. and Mohammad, A l i J . : On con verge nce of der i vat i ves of a ne w

sequence of l i near pos i t ive operators, Kyunpook Math. J . , S ubmi tted for pub l icati on . [ 2 ] Agrawal, P.N. and Thamer, K.J . : Linear combi n�tions of S úísz - B askakov type

operators, Demonstrat io Math. ,32(3) ( 1 999), 575-580. [ 3 ] Goldberg, S . and Meir, Y. : Minimum modul i of different ial operator, Proc. London

Math. Soc . , 23 ( 1 97 1 ) , 1 - 1 5 . [4] May, c.P . : S aturat ion and in verse theorem for combi n at ions for a c l ass of exponent ia l

type operators, Canad. J . Math. , 28 ( 1 976) , 1 224- 1 25 . [ 5 ] Rathore, R .K .S . , L inear combinat ions of l i near pos i t i ve operators and generat i n g

re l at ions i n spec i al funct ions , Ph.D. Thes is , I. I.T. De lh i , ( 1 97 3 ) .

Department of M athemat ics lnd ian lns t i tu te of Techno logy-Roorkee, Roorkee-247 667 , INDIA. E-mai l ' : pnappfm a @ i i tr .ernet . i n E-mai l * . : al ij asmoh @ yahoo.com

Recibido : 2 de Noviembre de 200 l . Aceptado : 1 de Julio de 2002.

Rev. Un. Mat. A lgentilla, Vol. 44- 1

REVISTA DE LA UNION MATEMATICA ARGENTINA Volumen 44, Número 1 , 2003 , Páginas 43-52

ON CONVERGENCE OF DERIVATIVES OF A NEW SEQUENCE OF LINEAR POSITIVE OPERA TORS

P.N. Agarwal and AIi J . Mohammad

43

ABSTRACT. In the present paper, we introduce a new sequence of l inear positive operators lo approximate a class of unbounded continuous functions of exponen ti al growth on the ¡nterval [O. co ) . First. we study the basic pointwise convergence theorem in s imultaneous

approx imation and then proceed to study the degree of this approximation .

l . INTRODUCTION Let C[O. co) denote the c1ass of all continuous functions on the interval [O . co) . For a > O and

f E Ca [O, co) =: {f E crO, co) : If(t) 1 s, M ea l for some M > O} , we define a sequence 01' l inear

posit ive operators Mn as

00 00 ( 1 . 1 ) M/7 c/(t) ; x) = nI P/7." (x) fq /7.1'-\ (t )f(t )dt + (l + x)-

n f(O) . 1'=1 o (n + v - lJ V -/7-1' e -ni (nt )"

where Pny (x) = X (l + x) , x E [O , co) , and qn. ,, (t ) = ,

t E [O, CO) . V v .

The space Ca [O, co) i s normed by I lfl l e = sup If(t ) l e -a l , f E Ca [O, co ) . Alternatively, the " 0:9<00

opcrator ( 1 . 1 ) may be

oc:

written as a:

Mn c/(t ) ; x ) = fW" Ct. x)f(t )dl , o

W" ( t . x) = nI Pny (x) qn,v- \ ( t ) + (l + x) -n o(t) , o(t ) being the Dirac-delta function. 1'= \

where

The obj ect of this paper is to study some direct results in simultaneous approximation 01' the operator ( 1 . 1 ) . Some important references for the study in this are a are [2-4] . The study in simultaneous approximation (the approximation of derivatives of function by the corresponding order derivatives 01' the operators) was initiated by Lorentz [6] , who establi shed the pointwise convergence theorem in simultaneous approximation for Bernstein polynomial s on [0, 1 ] . His method for the pointwise convergence in simultaneous approximation has been successively appl ied by several workers to other operators (cf. [ 1 J, [ 5 ] , [7] etc . ) .

KEY WORD S : Linear positive operators, S imultaneous approximation, Degree of approximation, Modulus 01' continuity.

Rev. Un. Mat. A rgentina, \Ir)!. 44-1

44 P. N. AGRAWAL AND AL! 1. MOHAMMAD

2. DEFINITIONS AND AUXILIARY RESULTS

Let NO denote the set of non-negative integers . For m E NO , let the m - th order moment of

the Lupas operators i s defined by f-ln,m (x) = �Pn .v (X) (� - x)m LEMMA 1 [7] . For the function f-ln,m (x) , we have f-ln,o (x) = 1 , f-ln . l (x) = O and there holds

the recurrence relation nf-ln,m+ 1 (x) = x(l + x) [f-l�,m (x) + mf-ln,m-I (x)] , for m � 1 . Consequently, we have

( i ) f-ln.m (x ) is a polynomial in x of degree at most m; ( ii ) for every x E [0, 00) , f-ln,m (x) = O(n-[(m+ I ) / 2] ) where LB] denotes the integer part of fJ .

Let the m - th order moment ( m E N o ) for the operators ( 1 . 1 ) be defined by ro ro Tn.m (x) = M n ( ( t - x) m ; x) = n L Pn,v (x) Jqn,v-I (t ) (t - x) m dt + (_x)m (1 + x) -n . v=1 °

LEMMA 2. For the function Tn ,m (x) , there follow Tn,o (x) = 1 , Tn, 1 (x) = O and

( 1 .2 ) nTn,m+ 1 (x) = x(l + x)T�,m (x) + mTn,m (x) + mx(x + 2)Tn,m_1 (x), m � 1 . Further, we have the fol Iowing consequences of Tn ,m (x) :

( i ) Tn.m (x) is a polynomial in x of degree m, m -:f:. 1 ;

( ii) for every X E [O, OO) , Tn.m (x) = O(n-[(m+1 ) / 2 ] ) ; ( i ii ) the coefficients of n-k in Tn .2k (x) and Tn, 2k-1 (x) are el {x(x + 2 ) }k and

e2 x k- l (x + 2) k-2 (x 2 + 3x + 3) respectively, where el and e2 are some constants

dependent on k . Proof: It is easy to show that Tn , o (x) = 1 and Tn. l (x) = O . Next, we prove ( 1 .2) . For x = O , i t

clearly holds for alI m � 1 . For x E (0, 00) , we have

T�,m (x) = n f {p�,v (x) jqn,v-I (t ) ( t - x) m dt - mp n,V <x) jq n,v- I (t ){t - x) m- I dI) v=1 o O - m(-x)I7l-I (1 + x)-n - n (-x)m (l + x)-n-I .

Since x(l + x)p' (x) = (v - nx)p (x) and tq ' (t) = (v - nt)q (t ) , we have n,v n,v n,v n.v x( l + x)T;;.m (x)

."X; 00 = n L(v - nx)Pn.v (x) Jqn ,v-I (t )(t - x) m dt - mx(l + x)Tn,m_ 1 (x) + n( _x)m+ 1 (l + x)-n ,'= 1 o

.""1:.. 00 = n L Pn. v (x) J(V - nx)q n ,v-I (t ) (t - x)m dt - mx(l + x)Tn,m_1 (x) + n( _x)m+ 1 (l + x)-n v=1 O

Rev. Un. Mat. A rgentil1a. Vol. 44- 1

ON CONVERGENCE OF DERIVATIVES

00 00 00 00 = n ¿ Pny (x) f( v - 1 - nt + I )q ny-I (t)(t - x)m dt + n2 ¿ Pn,v (x) f q n, v-I (t)(t - x) m+ 1 dt v= 1 o v=1 o -mx(1 + x)Tn,m_1 (x) + n ( _x)m+1 (1 + x)-n

oc � 00 00 = n ¿ p¡¡y (x) ftq�y-I (t )(t - x) 1Il dI + n ¿ Pny (x) fqnY-1 ( / ) ( 1 - xr' dI + n Tn,l1l+ 1 (x) v= 1 o v=1 o

� 00 00 00 -mx(l +x)Tn,m-I (X)

= n ¿ Pny (x) fq�y-I (/ )(t - x)m+ 1 dI + xn ¿ Pn,v (x) fq�,v-I (t ) (I - x) m dt + Tn,m (x) 1 '= 1 o v=1 o - (_x)m (l + x)-n + nTn,m+ 1 (x) - mx(l + x)Tn,m_1 (x) ,

Integrating by parts, we get

x( l + x)T,; ,m (x) = -(m + i)Tn,m (x) + (m + 1)( -x) '" ( l + x)-11 - mxTn,Il1_1 (x) + mx ( _x)m-I (l + x)-n + T",m (x) - (_x)m (l + x)-n + nT",m+ 1 (x) - mx( 1 + x)Tn,m_1 (x ) ,

from which ( 1 , 2 ) is immediate ,

45

From the values of Tn,o (x) and T", 1 (x) , it is clear that the consequences ( i) and ( ii) hold for

m = O and m = 1 . The consequence ( i) can be proved easi ly by using ( 1 . 2) and the induction on m . We sketch below the proof of the consequence (ii) . Suppose that the consequence ( ii) be true for m, then by ( 1 .2 ) , we have

Then,

nTn,m+ 1 (x) = O(n-[(m+ l l / 2 ] ) + O(n-[(m+ I ) I 2] ) + O(n-[m l 2 ] ) = {o(n-[(m-l l / 2] ) , if m i s odd

O(n-[m / 2] ) , if m i s even

{ O(n-[ (m+ I ) / 2 1 ) i f m is odd Tn.Il1+ I (x) = O( -[(m+2l / 21 ) " f ' . n , 1 m IS even

Hence, for every x E [O, <Xl ) Tn m+1 (x) = O(n -[(m+2l l 2] ) . Thus, consequence (ii) holds for m + l . Consequently, by mathematical induction, it holds for all m E NO . The proof of consequence ( iii) fol lows easily from ( 1 .2) using mathematical induction on k and hence the detai ls are omitted I Our next result is a Lorentz-type lemma for the derivatives of the kernel Wn (l , x ) of the

operator M n .

LEMMA 3 [7] . There exist the polynomials qi,j ,r (x) independent of n and v such that

d '� [x v ( l + X) -n-v ] = ¿ni (v - nx) J qi,J,r (X) x v-r ( l + x) -n-v-r . dx 2i+j'Sr i,J�O

Rev. UIl. Mar. A1xentil1Cl. Vol. 44-1

46 P. N. AGRAWAL AND AL! J. MOHAMMAD

Corolla ry. Let c5 and r be any two positive real numbers and [a , b ] e (0, 00) . Then, for any m > ° we have, 00 sup n¿ Pn ,v (x) fqn,v_l ( t ) e Y t dt = O(n-m ) .

xe[a,b) v=1 I t-xl�o Making use of Taylor ' s expansion, Schwarz inequality for integration and then for summation and Lemma 2, the proof of the Corol lary easily fol lows, hence the detai l s are omitted.

3. MAIN RESUL TS First, we prove that the derivatives of the operator ( 1 . 1 ) are approximation processes for corresponding order derivatives of the function, i . e . , we prove that

M }r ) (f(t ) ; x) � f(r ) (x) , as n � 00 , r = 1 , 2, . . . .

THEOREM 1 . S uppose that r E N , f E Ca [0, 00) for some a > ° and f (r ) exists at a

point x E (0, 00) , then

( 3 . 1 ) n�oo

Further, if f( r ) exists and is continuous on (a - r¡, b + r¡) c (O, oo), f¡ > O , then (3 . 1 ) holds

uni formly in x E [a , b ] . Proof: By Taylor ' s expansion off , we have

l' f( i) (x) . f(t ) = ¿-.-, - (t - x) / + &(t, x)(t - x) 1' ,

i=O 1 . where &(t, x) � ° as t � x . Hence

00 Mn ( r ) (f(t ) ; x) = fWn (r ) (t , x)f(t)dt

o l' ¡ ( i ) ( ) 00 00

=:' ¿ . . , X fWn ( 1' ) (1 , x)(t - x) i dt + fWn ( 1' ) (1, x)&(t , x)(t - x( dt := 1 1 + 12 , / =0 /. o O 00

Using Lemma 2, we get that M n (1 m ; x) = JWn (t , x) t m dt , is a polynomial in x of degree

O exactly m, for all m E N ° . Further, we can write it as

M ( ni ) (n + m - I ) ! m ( 1 ) (n + m - 2) ! m- I OC -2 ) ( 3 . 2 ) n t ; x = x + m m - x + n ,

. n m (n - l ) ! nm (n - l ) ! and thus

1 , = ¿ � ¿ l. (_x) i-i -1' fWn (t , x) t idt = ,x �.+ r - . r ! l' ¡ ( i ) ) i ( '

J d I' [00 1 f(r ) ( ) [ ( 1 ) ' ] i=O l. i=O } dx O r . n (n - l ) !

= ( 1' ) (X)[ (n + r - 1) !] � f(r ) (x) , as n � oo . . n l' (n - l) !


Next, making use of Lemma 3 , we have 00

12 = fWn ( r ) u, x)c(t , x)(t - x) ' dt o

, q i l r (x) � , oof = ¿ n i r' " r nL.. Pn,v (x)(v - nx) J qn,v_ I (t )C(t, X)(t - x) ' dt 2i+j�r X ( 1 + x) v=1 O

i .)'?O

Therefore,

+ (- I{ (n + r - I ) ! ( 1 + x) -n-r c(O, x )(-x) " . (n - I ) ! , lq i J r (x)1 � I I/XJf I I I I r 1 / 2 1 � ¿ n i

r"

r nL.. Pn,v (x) v - nx qn,v- I (t ) c(l , x) t - x dI 2 1+ / �r X ( 1 + x) v=1 O

1 . j'? o (n + r - I ) ! -n-r I I r + ' (1 + x) c(O, x) x := /3 + /4 , (n - I ) !

47

Since c(t , x) � O as t � x , for a given c > O , there exists a 0 > 0 such that Ic(t , x)l < é' , whenever 0 < It - xl < O . For Ir - xl � O , there exist a constant K > O such that

IC( l , x)(t - x/ 1 � K ea l . Hence,

13 � CI ¿ n i+ 1 i >n,v (x) l v - nxl ) [ fqn,v-I (t )c lt - xl ' dt + fqn.v_ I ( I )K ea1 dl] := /5 + 16 , 21+ /�r v=1 I I-xl«> I '-xl�o i,j'?O , Iq i. ¡ ,r (x)1

where e 1 = sup , . 2i+)�r x r ( 1 + x/

l . j '?O Now, applying Schwarz inequality for integration and then for summation we are led to , 1 / 2 1 / 2 15 :s:; c C1 ¿ n i+ 1 i >n.v (x)lv - nx' j [lq n,v- I (t )dt] [1 q n ,v-I (t )(t - x) 2r dI] 2 1+ J�r v=1 O O

i,)�O

:S:; cC1 ¿ n i [f Pn,v (x)(v - nx)2J ]1 / 2 [n'f Pn,v (X>fqn'V_I (t )(t _ X) 2r dt] 1 / 2 21+ J�r v=1 v= 1 O

i ,)'?O From Lemma 1 , we have

� ( )2) _ 2j [� ( v ) 2 / -n 2 / ] � PI1. , , (X ) v - nx - n ;:OPn. v (x) -;; - x - (1 + x) (-x) (3 . 3 ) = n2) lO(n-) + O(n-S )J = O(n) (for any real s > O ) ,

Similarly, Lemma 2 yields us

48

'l_ Xl

P. N. AGRAWAL AND ALI J . MOHAMMAD '

n ¿ Pny ( X) fq 11 . 1' - 1 (t )(t - x) 2r dI = Tn,2r (x) - (1 + x) -n (_x) 2r I '� I () ( 3 .4) = O(n-r ) + O(n'-S ) = O(n-r ) (for any real .1' > 0 ) .

Therefore, 15 � & CI I n ' O(nj 1 2 ) O(n -r 1 2 ) = & 0(1 ) . 2 1+ J�r 1,)"20

00 . Next, we can write 16 = C2 In'+1 I Pn,v (x) lv - nxI J fqn ,v_ I (t ) ea 1 dl , where C2 = KCI . 21+ J�r v;1 I I-xl:;,o i ,j:;'O Hence, again using Sehwarz inequal ity for integration and then for summation, ( 3 . 3 ) and

Corol lary, we have 1 / 2 1 / 2 16 � C2 I ni+1 i >n,v (x) lv - nx1j [ fqn,V-I (t )dl] [ fqn,v_I (t ) e 2al dl] 2/+ /�r v;1 I I-xl:;,o I I-xl:;,o

i .j:;'()

� C2 1 I ni [I Pn,v (X)(V - nx )2J ] I / 2 [n I Pn,v (x) fq�,v-I (t ) e 2al dl]I / 2

_ /+ J�/ v;1 v=1 I I -xl:;,o / .}:;'O = O(n ( r l 2 )-.\· ) = 0 (1 ) (for s > r / 2 ) .

Now, sinee & > O is arbitrary, it fol lows that 1) � O as n � oo . Also, 14 � O as n � 00 and

henee 12 = o( l ) . Combining the estimates 01' / 1 and 12 , (3 . 1 ) is immediate .

The uniformity assertion fol lows easily from the faet that in the aboye proof 0(&) can be

chosen to be independent 01' x E [a, b] and all the other estimates hold uniformly in x E [ (1 , b] I Our next result is a Voronoskaja type asymptotic formula for the operators M n ( r ) ( f; x ) .

THEOREM 2. Let f E Ca [0, 00) for sorne a > O . If f( r+2) exists at a point x E ( 0, 00) , then

lim n ( MI1 ( r ) (f(l) ; x) - f (r ) (x» ) n�'x:;

( 3 . 5 )

Further, i f /( 1'+2 ) exists and is eontinuous on the interval «(1 - 1] , h + 1]) e (0 , 00) , 1] > O , then

( 3 . 5 ) holds uniformly on [(I, b] .

Proof: By the Taylor' s expansion 01' / , we get

1'+2 fU ) ( ) 00 00 M,, ( r ) (f (t ) ; x) = I-' -'I-x- fWIl ( r ) (t , x)(I - x) i dl + fWn ( r ) (I , X)&(t , x)(t _ x) r+2 dl i=() 1 . () O := 1 1 + 12 ' where &(1 , x) � O as I � x .

By Lemma 2 and (3 .2) , we have


= LJ!l (n ,+ r ) ' r ! + x (r + I )(-x) fWn(r ) (t , x) t r dt + fWn( r ) ( t , x) t r+ ' dt ( r ) [ - 1 ' ] j(r+ l ) ( ) [ 00 00 1 r ! n J (n - I ) ! (r + I ) ! o o

+ (�) (r + (r + ) (_x) 2 fWn(r ) (t , x) t rdt + (r + 2)(-x) fWn ,r ) (t , X) l r+ l dl j' ( r+2 ) [ 1 ) 2 00 00

(r + 2) . 2 o o

+ ÍW"" ' (/ , X) I"2dl 1 ' I r ) [ e n + r - I ) ! l j(r+ l ) (X) [( 1 )( )( (n + r - I ) ! '1 r (n + r ) ! ( 1 ) ' = f (x) r + r + -x r r . + r+1 r + . x n (n - I ) ! J (r + l ) ! ' n (n - l ) ! J \ n (n - l ) !

( 1 ) ' (n + r - I) ! J j(r+2 ) (x) [ (r + I )(r + 2) 2 ( n + r - I ) ! 'J + r + . r + x r . nr+ 1 (n - I ) ! (r + 2) ! 2 n r (n - 1 ) ! ( (n + r ) ' (n + r - I ) ' J + ( r + 2)(-x) 1 . ( r + I ) ! x + (r + I ) ! r I .

nr+ (n - I ) ! nr+ (n - 1 ) ! ( (n + r + I ) ! ( r + 2) ! 2 ( 2)( 1 ) (n + r) ! ( 1 ) ' J l O -2 + 2 X + r + r + 2 r + . x + (n ) . n r+ (n - I ) ! 2 nr+ (n - 1 ) ! = j ( r ) (X)[ (n + r - I ) ! ] + ¡<r+ l ) (X)[ r (l + x) (n + r - 1 ) !] , n r (n - I ) ! n nr (n - 1) !

+ j ( r+2 ) (X)[ nx 2 + 2nx + rx 2 + 2r 2 x + 2rx (n + r - 1) !] + O(n -2 ) . 2n 2 n r (n - 1 ) !

49

Hence in arder to prove (3 . 5 ) it suffices to show that n/2 � ° as n � ro , which fol lows on

proceeding along the l ines of proof . of /2 � ° as n � ro in Theorem l . The uniformity

assertion fol lows as in the proof of Theorem 1 I Now, we present a theorem, whi ch gives an estimate of the degree of approximation by M,, ( r ) ( . ; x ) far smooth functions.

THEOREM 3. Let j E ea [O, ro) for some a > ° and r :S; q :S; r + 2 . If j(q ) exists and is

continuous on (a - '7, b + '7) e (O, ro) , '7 > 0 , then for sufficiently large n, ! !Mn ( r ) (f(t) ; x) - j(r ) (X)! ! :s; e l n - 1 I l lj(i ) 1 1 + e2 n -1 / 2 OJ ¡l r+ ' ) (n - 1 / 2 ) + O(n -2 ) , i=r

where el ' e2 are both independent of f and n , OJI (5) is the modulus of continuity of j on

(a - '7, b + '7) . and 1 1 . 1 1 means the sup-norm on [a, b ] .

Rev. UIl. Mat. A rgentina. Vol. 44-1

50 P. N. AGRAWAL AND AL! J . MOHAMMAD

Proof: By our hypothesis

q f ( i ) (x) f(q ) (J:) f(q) (x) . f(t ) = ¿-.-, - (1 - x) ; + ." -, (t - x)q x(t) + h(t, x)(I - x(1» , ;=0 / . q.

where ; lies between t, x, and X(t) is the characteristic function of the interval (a - "1, b + "1) . For t E (a - "1, b + "1) and x e [a, b] , we get , q fU) ( ) f(q ) (J:) f(q ) ( ) l(t) = ,, __ x_ (t - x) ; + ." - x (t - x) q . . � " , ;=0 1. q . ' For l E [0, 00) \ ( a - 1], b + 1]) and x E [tz, b] , we define

Now,

q f ( i ) (x) h(t, x) = f(t ) - "--(t - x) i . � " ;=0 l .

[ q fU) ( ) 00 1 M,, ( r ) U(t ) ; x ) - f (r ) (x) = � --+ JWn (r ) (t , x)(t - x/ dt - f (r ) (x)

+ Jw/' ) (t , x) ) -, x) (t - x)q x(t ) dt + JWn ( 1' ) (t , x)h(t , x)(1 - x(1» dt 00 {f (q ) (; f(q ) ( } 00 o q. o .

:= /1 + 12 + /3 ' By using Lemma 2 and (3 ,2) , we get

1, = ¿ � ¿ l. (-x) ;-j -1' JWn (t, x) t j dt - f(r ) (x) q f ( i ) i ( '

J d I' [00 1 ¡= () 1. j=o .l dx (}

= ± f( i� (x) ±( i.)( _X) i-j �[(n + j - 1) ! xj + j(j - 1) (n -: j - 2) ! x j-I + O(n-2 )] _ f(r ) (x) . i=r I! j=r ] dxr nJ (n - 1) ! nJ (n - 1 ) ! Consequently, I l/dl � el n -I [� llf ( i) I I) + O(n-2 ) , uniformly in x E [a, b] . To estimate /2 we proceed as follows :

00 { I l (q ) (;) - fq ) (x)1 } 1 /2 1 � JI W/r ) (t, X) 1 . , It - xl q X(t) dt o q .

á1f( q ) (o) 00 1 1 ( It - xl ] .� q ! J Wn ( r ) (t , x) l + -o- lt - xl q dt

< "'1';, (o) [ n � 1 p<�" (x) 1 1 q ,_"_ , (1 )( 1 1 - xl ' + 0-'1' - xl'· ' ) dI

Now, for s = 0, 1, 2, . . . , we have

Rev. Un. Mat. A rgentilJa. Vol. 44- 1

+ (n + r - 1) ! (1 + x) -n-r ( Ixl q + o- l lx l q+ 1 )l, o > O . (n - l) ! J

x OC) n¿ Pn.v (x) lv - nx Y fqn.v_ l (t) lt - xI S dt v�1 o

� n �p"", (x) Iv - mf [(1q"",-, (I)dl ]"'[]q" ",-, (1)(1 - x) " dI r 1 � [� p"", (x)(v - nx)" f' [n � p"", (X)]q " ",-J (t)(t - x) " dI J" = O(nj l 2 )O(n-s / 2 ) = O(n (J-s ) / 2 ) ,

uniformly in x E [a, b] , in view of (3 . 3 ) and (3 .4) . Therefore by Lemma 3 , we get

J.. 00 n¿ l pn)r ) (X ) 1 fqn .v- I ( t ) 1 1 _ xl s dI v� 1 O

� n f ¿ ni lv - nxl j l�i.j .r (X)l Pn.v (x) Iqn,v-I (t ) lt - xl s dt

v=1 2i+jsr X (l + x) O i.j�O

� \;�,�; [n �p"", (x) lv - nxl l ]q"",-, (1) 1' - xl ' dI] � K u�,� ; O(n 1h' ; ' )

l .j�O ; .j�O ( 3 . 6 ) = O(n ( r-s ) / 2 ) ,

5 1

Iq r (x) 1 uniformly in x E [a. b }, where K = sup sup , l .} , . Choosing 0 = n"': 1 / 2 and applying

2;+ jsr xE[a.b] x ' ( l + x) r i .j�O

(3 . 6 ) . we are led to ( - 1 / 2 ) (¡) '« 1 ) n [ ] 1 1 /2 1 1 � j I

O(n(r-q) / 2 ) + n I / 20(n (r-'1-1 ) / 2 ) + O(n -m ) , for any m > 0 q.

< C n -(r-q ) / 2 úJ (n - 1 I 2 ) - 2 f� ' Since f E [O, oo) \ (a - lJ, b + IJ) , we can choose 0 > 0 in such a way that It - xl ;::: o for all

x E [a.b] . Thus, by Lemma 3 , we obtain

I I � ,, ; I I j Iqi,j .r (x) 1 f I I /3 � nL..., L..., n v - nx r r Pn.v (x) qn.v_l (t) h(t, x) dt v=1 2i+ jsr X (l + x) 1 !-XI�8

i.j�O + (n + r - l) ! ( l + x)-n-r lh(o, x)l · (n - I ) !

For l' - xl ;::: o . we can find a constant M > O such that I h(t , x)1 � M ea! . Finally using Schwarz

'inequal ity for integration and then for summation, (3 . 3 ) , and Corol lary , it eas i ly follows that

52 P. N. AGRAWAL AND ALI J . MOHAMMAD

13 = O(n-s ) for any S > O , uniformly on [a, b] . Combining the estimates of 11 ' /2 ' /3 , the

required result is immediate I

ACKNOWLGEMENT The authors are extremely thankful to the referee for his valuable comments.

REFERENCES [ 1 ] Agrawal, P .N . and Thamer, Kareem J . : Linear combination of Szasz - Baskakov type

operators, Demonstratio Math. 32 ( 1 999), 575-580 .

[2] Derriennic, M .M. : Sur I ' approximation de functions integrable sur [0, 1 ] par der polynomes

de Bernstein modifies, 1 . Approx. Theory 31 ( 1 98 1 ) , 325-343 . [ 3 ] Ditzian, Z. and Ivanov, K . : Bernstein-type operators and their derivatives, 1 .

Approx.Theory 5 6 ( 1 989),72-90. [4] Oonska. H.H. and Zhou,X. -L . : A global inverse theorem on simultaneous approximation by

Bernstein-Durrmeyer operators, J . Approx. Theory 67 ( 1 99 1 ) , 284-302. [ 5 ] Kasana, H. S . and Agrawal, P . N. : Approximation by l inear combination of

Szasz - Mirakian operators, Colloq. Math. 80 ( 1 999), 1 23 - 1 30 . [ 6 ] Lorentz, 0 . 0 . : Bernstein Polynomials, Univ. Toronto Press, Toronto , 1 95 3 . [ 7 ] Sinha, R .P . , Agrawal, P .N . and Oupta, Vijay : On simultaneous approximation b y modified

Baskakov operators. Bul \ . SOC . Math . Belg. Oen. Ser. B. 43 ( 1 99 1 ) , 2 1 7-23 1 .

Department of Mathematics lndian Institute of Technology-Roorkee, Roorkee-247 667, INDIA. Email

*: [email protected] . in

Email* *

: al ijadma@isc . i itr.ernet. in

Recibido : 19 de Junio de 2002. Aceptado : 1 3 de Febrero de 2003 .



Spline wavelets in periodic Sobolev spaces and application to high order collocation methods

F. Bastin , C . Boigelot , P. Laubin

Abstract In this paper , we present a particular family of spline wavelets constructed

from the Chui-Wang Riesz basis of L2 (1R) . The construction is explicit, al

lowing the study of specific functional properties and rather easy handling in

numerical computations. This family constitutes a Riesz hierarchical basis in

periodic Sobolev spaces . We also present a necessary and sufficient condition

of strong ellipticity for pseudodifferential operators obtained with respect to

these splines . It uses a new expression for the numerical symbol of the bound

ary integral operators . This expression allows us to use efficiently collocation

methods with different meshes and splines .

Key words. Sobolev spaces, splines, wavelets, collocation methods.

2000 Mathematics Subject Classification: 46E35, 65N35 .

1 Introduction

53

Collocation methods using splines is a natural and widely used technique for solving strongly elliptic pseudodifferential equations on closed curves (see [2] ' [8] , [20] ) . However, stability, asymptotic convergence, good condition numbers and efficient compression are not so easy to obtain. Por smooth boundaries , the convergence of these methods has been proved by Arnold , Saranen and Wendland ( [1 ] , [2] ' [ 19] ) . Several recent papers use these methods i n a more general setting (see for example [ 1 2] ' [ 14] , [ 1 5] ) .

Wavelets can b e used in this context because they provide Riesz bases, allow progressive computations and give good compression schemes . Moreover, one can adapt the construction of the basis so that the properties of the functions solve or reduce technical and numerical difficulties .

In this paper, we first present new explicit constructions of Riesz bases of spline wavelets in periodic Sobolev spaces based on the Chui-Wang wavelets and we study typical properties of these bases . We focus on spline functions especially because they are easy to handle in implementation. We also show how to obtain the dual bases explicitly.

Then we discuss the usefulness of these bases to obtain good convergence and asymptotic stability for collocation methods in the resolution of boundary integral equations . In this framework , we present a result concerning the numerical symbol


54 F. BASTIN, C . BOIGELOT AND P. LAUBIN

of periodic pseudodifferential operators . We give a proof of the characterization of the coercivity condition which leads to relations on the mesh es and order of the splines that is easy to handle (see Theorem 7) .

As a typical example of application, we treat the simple and double layer potentials for the Dirichlet problem of the Laplace operator . The first one involves Sobolev spaces of half integer orders and the second one Sobolev space of integer orders . Some numerical computations of the condition number are presented and confirm the theoretical results .

2 Spline wavelets in perio dic Sobolev spaces

2 . 1 ' Chui-Wang wavelets

For any strictly positive integer m and any integer j , denote by Vj(m) the set of functions on lR which are smooth splines of degree m - 1 with respect to the mesh {2-j k : k E Z} and belong to L2 (lR) . If Ó E [0 , 1 [ , we denote by Vj�';') the same set of Hplines but with respect to the mesh { 2-j ( k + Ó) : k E Z} . The corresponding sets of 1-periodic splines are respectively denoted by VJm) and VJ:;) .

Let Nm = X[O,l ] * . . . * X[O,l ] , " v m

be the cardinal spline function . The classical Chui-Wang spline biwavelet 'l/;m E V1(m) iti defilled by

with

+00 m- l ¿ I Nm (� + 2k7rW = ¿ e-ik{ N2m (m + k) . k=-oo k=-m+ l

It is well known (see [6] ) , that the functions 'l/Jm(x-k ) , k E Z, form a Riesz basis of the orthogonal complement WJm) of Vo(m) in V1(m) . Moreover supp( 'l/Jm ) c . [O , 2m - 1] , Sillce we have orthogonality between the levels, the functions

form a Riesz basis of L2 (lR) . Let

and


' > O J -

SPLINE WAVELETS IN PERIODIC SOBOLEV SPACES 55

1 t is readily seen that for every j, the functions W m;j,k , ° ::; k < 2j , form a Riesz basis of the orthogonal complement of V;ml in vj�i and that we have orthogonality between the levels . The Riesz bounds are independent of j . The spline functions 1 and Wm;j,k , j 2=: O, ° ::; k < 2j , form a Riesz basis of L2 ( ]0 , I D . Moreover, after normalization of the constant function 1 , the Riesz bounds in L 2 ( ] O , 1 [) are the same as the ones obtained for the functions Wm;j,k , j, k E Z, in L2 (lR) .

Let us consider the Sobolev spaces H;er(]O , 1 [) , s E lR, of l-periodic distributions endowed with the norm

I l u l l ks = l uo l 2 + l u lks pcr pcr where

l u l kí�cr = 2:= Ik l 2s I uk l 2 k¡O

and 'Uk is the k-th Fourier coefficient of u. We also use the notation

I l u l l o = I l u l l L2 (]O, l [) ' We recall also that in case s > 1/2 , the norm 1 1 , I I Hrcr is equivalent to the following one

I l u l l ; = lu (OW + l u lkpcr = lu (0 ) 1 2 + 2:= Ik l 2s l uk l 2 . k¡O

The stability of Chui-Wang wavelets in Sobolev spaces has been studied in [ 16] and [ 10] . This result is covered in greater generality in various studies . However , for the sake of completeness , we give here a simple and very direct proof in the periodic setting with optimal indices .

Proposition 1 J! 1 8 1 < m - � then there are e, C > O such tha t +00 2j - 1 +00 21 - 1 +00 2j - 1

C 2:= 2:= I Cj,k 1 2 ::; 1 1 2:= 2:= cj,kTjSWm;j,k l l kpcr ::; e 2:= 2:= I Cj,k I 2 . j=O k=O j=O k=O j=O k=O

This result is optimal since Wm;j E H;er (] O , 1 [) if and only if s < m - � . Proof Let

21 - 1 Qj! = Pj+1! - Pj! = 2:= CjkWm;j,k .

k=O Here Pj is the L2-orthogonal projection onto vjml . Let s be such that I s l < m - � . We have

I I Qj! I I Hpcr ::; e2js l l Qj! 1 1 0 Vj E N.

lndeed, i f 8 2=: O , this property follows from the inverse property of periodic splines in Sobolev spaces ( see for example , Theorem 2 . 1 1 of [ 17] ) . If s < 0, we write

Rev. Un. Mat. Argentina, Vo!. 44- 1

using the approximation property of splines (see Theorem 2 . 6 of [ 1 7] ) . Now, if E > ° satisfies 1 8 ± E l < m - � , we get

+00 +00 1 1 ¿ TjsQj f l l �pcr < L 2-2jS I I Qjf l l �pcr + 2at L T(j+k)s (Qjf , QkJ) Hpcr j=O

Finally +00

j=O j<k +00 < e L I I Qjf l 1 6 + 2at ¿ T(j+k)s (Qjf , Qk J) Hpcr j=O j<k +00 < e ¿ I I Qjf l 1 6 + 2 ¿ T(j+k)S I I Qj f I I H�;t: I I Qkf I I H�;;-: j=O j<k +00 < e L 2- l k-j l € I I Qjf I I 0 1 IQd I I 0 j,k=O

+00 < el ¿ I I Qjf I 1 6 ·

j=O

¿ I I Qjf I 1 6 = /� 2jSQjf , � TjSQjf) \ j=O j=O L2 (]0 , l [) j=O +00 +00

< 1 1 ¿ 2jsQjf I IH;c� 1 1 ¿ TjsQj f l l Hpcr j=O j=O +00 +00

< e ¿ I I Qjf l 1 5 1 1 ¿ TjsQjf l l Hpcr ' j=O j=O This proves the proposition. o

2 . 2 Modified spline wavelets

Since , by construction , the function 'ljJm has m vallishillg momellts, there is a unique spline function (jm E Vp

m) on IR such that DmOm = 'I/Jm and supp(Om ) e [O , 2m - 1 ] , Explicitly, we have

and also

Let +00 8m;j (x) = Tmj+j/2 ¿ Om (2j (x - k) ) , j 2: O .

k=-oo Rev. Un. Mat. A rgentina, Vol. 44-1

SPLINE WAVELETS IN PERIODIC SOBOLEV SPACES

It foIlows that +00 supp (8m;j ) e U [k, k + Tj (2m - 1 ) ] .

k=-oo

57

In [0 , 1] , this set is reduced to an interval with length 2-j (2m - 1) if j is large . With the previous notation, we get Dm8m;j = \[J m;j ' We also consider the functions

These functions are not orthogonal to constants since ()m has no vanishing moment o But this property is "replaced" by the fact that they aH vanish at 0, which is very useful in the sequel . We give now the proof of this property.

Lemma 2 Por every m E N and every integer p, we have

It follows that for every m E N, j � 0 , k E {O , . . . , 2j - 1 } , and q E Z, we have 8m;j, k (2-j q) = O.

Prooj. Since +00

8m;j,k (Tjq) = Tmj+j/2 L ()m (q - k - 2j Z ) l=-oo

it suffices to show that

From the relations

we get

Pm (ONm (O = Tme-i(m- l )';Wm (� + n) (iOm N2m (O ;¡;m (O ( i�)m m- l +00 L e-ik'; N2m (m + k) = L e-ik'; N2m (m + k)

k=-m+l k=-oo

+00 T2me-i(m- l )'; L (_ l ) ke-ik'; N2m (m + k )N2m (�)

k=-oo +00

()m (x ) T2m+1 L ( _ l ) k N2m (m + k)N2m (2x - (k + m) + 1 ) . k=-oo


58 F. BASTIN, C . BOIGELOT A N O P. LAUBIN

lt foHows that for any integer p, if one changes the sum over k to a sum over k' with k + m = 2p - (m + k') + 1 , one gets

o

+00 e", (p) = T2m+ 1 L ( _ l ) k N2m (m + k) Nzm (2p - (m + k) + 1 )

k=-oo +00

- 2-2m+l L (- I ) k ' N2m (2p - (m + k') + 1 ) N2m (m + k' ) k'=-oo

Proposition 3 For any integer m 2: 1 and any real number s such that .� < s <

2m - � , the functions 1 and 2j (m-s)8m;j,k , j 2: O, O � k < 2j , form a Riesz basis of H;er ( ]O , I [) .

Proof. By construction , we have 1 (8m;j,k ) t ! = 1 ( 27rl ) -m (Wm;j,k ) 1 1 for l E Z, l f a hence

+00 2.1 - 1 I '\"' '\"' c 2J (m-s)8 . I ' L.-t L.-t ] ,k m;) ,k Hpcr j=O k=O

+00 2.7 - 1 ( 27r(m l L L Cj,k2j (m-s) Wm;j,k I Hg�,m

j=O k=O +00 2J - l

(27r(m l l L L Cj,k2j (m-s) Wm ;j,k I I H��,m j=O k=O

with 18 - mi < m - 1/2 . Using Proposition 1 , we then get constants e, e > O such that

+00 2j - 1 +00 2j -1 +00 2j - l

e L L ICj,k 1 2 � I L L Cj,k2j (m-s)8m;j,k l��c, � e L L I Cj,k I2 .

j=O k=O j=O k=O j=O k=O

Now, since the functions 8m;j,k aH vanish at O , we get

2 +00 2j - l = l eo l 2 + I L L cj,k2j (m-s) 8m;j,k l��c, '

j=O k=O

Using these two relations , we obtain that the functions 1 and 2j (m-s)8m;j,k , j 2: O , O ::; k < 2) , form a Riesz family.

The functions 8m;j,k are l-periodic spline functions of degree 2m - 1 with respect to the mesh { k2-j- 1 : k E Z} . The space V;!7) of these functions has the dimension 2)+ 1 and we have exactly 1 + 1 + 2 + . . . + 2j = 2)+1 elements of this space among the Riesz family. S ince the union of these spaces is dense in H;er (]O , ID , this proves t.he proposition. O

Rn: Un. Mat. A/;r?elltiná. Vol. 44- 1

It is now natural to ask for a characterization of the dual Riesz basis of the family

{ l } U { 2j (m-s) e · ' k J' > O O < k < 2j } m,) , 1 _ , _ ,

at least for S = m. If one remembers the link with the Chui-Wang wavelets , it is clear that this basis should be related to the dual of the Chui-Wang wavelets and

. constructed in the same way.

For any m E N, it is known that the function Wm , called the dual Chui-Wang wavelet , defined as

� �m(�) Wm (�)

= Lt:-.oo I�m (� + 2kn ) 1 2

has the follbwing properties (see [6] , [7] ) .

Proposition 4 For every j , k E 2 we set ;¡;m;j,k (X) = 2j/2;¡;m (2jx - k) , x E IR. 1) We have

+ 00 � � � L I Wm (� + 2h ) 1 2 = Wm (OWm ( -2 )wm (n + 2 ) ' � E IR. k=-oo

Ji follows that Wm is exponentially decreasing and that there are Ckm) , k E 2 and rm > O such that

+00 �}m (x) = L Ckm)Wm (X - k) , x E IR,

k=-oo sup eTm lkl l ckm) I < +00. k

2) For every j E 2, the family ;¡;m;j,k (k E 2) is the L2 (IR) dual of the Riesz basis Wm;j,k ( k E 2) of Wlm) . Sin ce the spaces W}m) are L2 -orthogonal to each other, we get that the family Wm;j,k (j , k E 2) is the dual Riesz basis Of Wm ;j ,k (j , k E 2) . 0

We define em as follows:

em (x) = ( � ) 1 r (x - t)m- l;¡;m (t) dt , x E IR. m I . J-oo Here again , it is readily seen that this function has the following properties .

Proposition 5 We have Dmem = ;¡;m on IR and em (x) = Lt:-.oo ckm) em (x - k) for every x E IR. Jt follows that em belongs to V?m) and is exponentially decreasing. O

For O � j and O � k < 2j , we also define

+00 . e=m',J' (X ) = 2-mj+j/2 """' e-m (2j (x - k) ) , e- ( ) e- ( k2-j ) L....t - m ;j ,k X = - m;j X - ,

k=-oo x E IR.

60 F. BASTIN, C. BOIGELOT AND Po LAUBIN

We see that these functions are continuous 1 -periodic functions and that they are related to the functions 8m;j,p , (j � O, O ::; p ::; 2j ) by the following relations

with

+00 2;- 1 8m;j (x) = 2: cim)8m;j (x - Tjk) = 2: Cj;')8m;j,p (X)

k=-oo p=O

+00 cj;,) = 2: C�:�k ' p = O, o o o 2j - l o k=-oo

Proposition 6 For any integer m � 1, the family { 1 } U { (27r)2m8m;j,k , j � O , O ::; k < 2j } , is the dual of the Riesz basis { 1 } U {8m;j,k , j � O , O ::; k < 2j } relatively to the sea lar produet

(f, g)m = f (O)g(O) + 2: I k l 2m /kgk o kfO

Proof. We only have to show the orthonormality between these two families o We have

and

(8m;j,k ' 8m;j' ,k/ ) m = 8m;j,k (0)8m;j' ,kl (0) + (27r) -2m (Wm;j,k ' �m;jl ,k/ ) O = (27r)-2m (Wm;j,k ' �m;jl ,k) O = (27r) -2mÓj,jI Ok,kl

(8m;j,k , l )m = 0 0 Moreover, using 8m(l ) = O ( l E Z) , we get

2L1 ,8m;j,k (0) = 2: Cj;')8m;j,p (-Tjk) = O

p=O for every j � O, O ::; k < 2j o Hence

( 1 , 8m;j,k)m = o. o

Remarko Other hierarchical Riesz bases of splines could be used for test and trial functions (the Chui-Wang periodic wavelets are the first example) o To get collocation methods, test functions have to be splines of degree 2m - 1 but for the trial functions , according to Theorem 7, we could use splines of any degreeo For example , we could also consider functions coming from l < m primitivations of 'lj;m Le o the functions Sm;l defined by Dl Sm;l = 'lj;mo These functions have m - l vanishing moments and the corresponding periodic functions

+00 Sm;l ;j (X) = 2-jl+j/2 2: Sm;I (2j (x - p) )

p=-oo Revo Uno Mato A rgentina, Vol. 44�1

SPLINE WAVELETS IN PERIODIC SOBOLEV SPACES 6 1

are such that 1 , 2j ( l-sl Sm; l ;j,k , j � O, O :::; k < 2j ,

form a Riesz basis of smooth 1-periodic splines of degre m + l - 1 for H;er (] O , 1 [) if l , s satisfy 1 /2 - m + l < s < m + l - 1/2 . This result is directly obtained using Proposition 1 .

Moreover , since the functions Sm;l have at least one vanishing moment , the functions Sm;l ;j are L2 (] 0 , I D orthogonal to constants. This leads to the fact that we see immediately that the dual Riesz basis of the family

1 , 2j ( l-sl Sm;l ;j,k , j � O , O :::; k < 2j , relatively to the scalar product

(j, g) Hbcr = fogo + L I k l 21 fkgk k,pO

is obtained by the same procedure as before , but with l primitivations of the dual Chui-Wang wavelet .

3 C ollocation with spline wavelets

3 . 1 Presentation of the problem

Let o be a smooth bounded and connected open subset of JR2 whose boundary ao is also connected. To solve the Dirichlet problem { -6u = O in O

U18!1 = f two methods are widely used. First , we can use the single layer potential representation of u

u(x) = - -21 r v (y) log I x - y l dO'(y) , x E O 7r J8!1

where the boundary integral equation for v is simply V v = f with

(Vv) (x)' = - � r v (y) log I x - y l dO' (y) , x E ao. 27r J 8!1 We can also represent the solution as a double layer potential

1 1 (x - y) .vy ( ) ( ) u(x) = -2 I 1 2 w y da y , x E O,

7r 8!1 X - Y where v is the unit inward normal to the boundary and the boundary equation for w is ( � + K)w = f with

1 1 (x - y) . vy ( ) ( ) Kw(x) = -I 1 2 W y da y , x E ao.

27r 8!1 X - Y It is known that V : HS (aO) --; Hs+1 (aO) is an isomorphism for every s if and

only if the analytic capacity of O is not 1 . Moreover � + K : HS (aO) --; HS (an) is an isomorphism for every s E IR. Since the boundary is smooth , K is a compact operator from HS (aO) to HS (aO) .

62 F. BASTIN, C. BOIGELOT AND P. LAUBIN

3 . 2 A result of coercivity

These two boundary operators are particular cases of the classical pseudodifferential operators with constant coefficients in the periodic setting :

A = b+Q! + b_Q� + Ko where b+ , b_ E e, (3 E IR, .

Q!u(x) = L lk l13uke2ik7rx , k'¡'O

Q�u(x) = L sgn(k) l k l13uke2ik7rx k'¡'O

and, for aH r E IR, Ko is compact from H�er into H�;/ . Indeed, if we use a parameterization of en proportional to arc length and defined in [O, 1 ] ' the boundary operators considered are of this form with : - (3 = O and b_ = O , b+ = � , Kou = T + Ku for the double layer potential - (3 = - 1 and L = O, b+ = 4� ' Ko a compact operator from H�er (] O , 1 [) into C�r for the single layer potential .

The foHowing result gives an estimate o f Céa type for strongly elliptic constant coefficients pseudodifferential operators . It concerns splines of any order (see also [ 1 7] ) . The technique used here presents a new express ion of the condition leading to the estimate of Céa type . This expression gives then an easy description of the relations between the degree (r) of the splines and the meshes (see 1 ) and 2) of Theorem 7 below) . It also leads to results on boundedness of condition number arising in numerical computations.

Theorem 7 Let m be a strictly positive integer, b E [0 , 1 [,

Au = b+Q!u + LQ�u + Uo a pseudodifferential operator and assume that s > � and r + � > s + (3 . Then there is c > O such that

sup I (Af , g)Hm 1 2 c l l f l lw+il (2m) per pcr 9EVj . lIgll H2m-s =1 pe,

for every j and every f E V;�+l ) if and only if b+ =1 ±b_ and the function

(r) ( B) 1+00 tr-13 N(r) (t , a , B) d a I-t ()" A a , = t o cosh t - cos B

does not vanish in ] 0 , 1 [ with B = 21Tb . Here

N(r) ( t , a , é) = ( b+ + b_ ) e-at (et - eiO ) + (- lt+l (b+ - b_ ) eat (ei6 - e-t ) . Defining c+ : = b+ + b_ , e := b+ --,. b_ and 'Y := inf{R.c+ , R.c_ } , we get the

following particular cases of application. 1) Assume (b- = O and b+ =1 O) or (b+ , b_ E IR and 'Y > O) . Jf r is odd (respectively r is even), the condition is satisfied if and only if b =1 � (respectively b =1 O) . 2) Assume 'Y > O . Jf r is odd (r:espectively r is even), the condition is satisfied in case b = O (respectively b = V.

Rev. Un. Mat. A rgentina. Vol. 44-1


Proaf. Let N = 21 . The Fourier coefficients

of 9 E V?m) satisfy

For j E vtt1) , we have

where the coefficients

satisfy

Using this, we get

Ck = 11 g(x) e-2i7rkx dx

+00 j (x ) = L ake-2ik7rÓ/N e2ik7rx

k=-oo

11 8 . ak = j (x + -)e-2mkx dx o N

+00 (Aj , g) Hm per aoco + ¿ I k I 2m+i3 (b+ + b_ sgn(k) )akcke-2ik7rÓ/N

k=-oo N- 1

aoco + L k2m+r+ 1akcke-2ik7ró/N dk k=l

which leads to

where, for k = 1, . . . , N - 1

d � -2i7rPÓ (b b (k N) ) I k + pNIi3 k = � e + + _ sgn + p (k + pN)r+ 1 '

p= - oo

and go E V(�m) is defined by J ,

+00 1 Pk = ¿ ( k + pN) 2s p= - oo

63


In the same way

+00 I l f l l�gt! = l ao l 2 + L I k I 2(s+m lak I 2

k=-oo N-1 +00 1 1 1 2 + � k2(r+1 ) 1 1 2 � ao . � ak p�oo I k + pN I 2(r+ 1) -2 (s+,6) .

It follows that the stated bound holds if and only if there is e > ° such that

� 1 < I dk l 2 e L; I k + pNI 2(r+ 1 )-2(s+,6) - Pk p=-oo for al1y N, k such that 1 :::; k <: N. Sil1ce both sides are homogeneous of degree 2(r + 1 - s - (3) with respect to (k , N) , this is equivalel1t to the existel1ce of e > ° such that

+00 1 +00 1

e L (p + a)2s L (q + a)2 (r+1 )-2(s+m p=-oo q=-oo +00

1 1 ,6 < 1 � e-2ip1r6 (b + b_ sgl1(p + a) ) p + a 1 2 ( * ) - L; + (p + a)r+ 1 p=-oo

for al1y a EJO , 1 [ . Lemma 8 below shows that , for a E J O , 1 [ , we have

The left hal1d side of ( * ) is a continuous and strictly positive function of a E ] 0 , 1 [ ; moreover it has the behaviour of a-2(r+1-m (resp. ( 1 - a) -2(r+ 1-,6» ) when a � 0+ ( resp . 1 - ) . U sing the results of Lemma 8, the assertion readily follows . O

To simplify notations in the following lemma, we assume that we are in the conditions of Theorem 7 and we set

e (a , e , r ) . -

where


- 1 -ip8 L (a, e , r ) = L ( _p � a) r+ 1-f3 ' p=-oo

Lemma 8 We consider the functian a E ] 0 , 1 [ 1---> E(a , e, r ) . 1) This function is continuous, satisfies

and

1 ¡+oo tr-{3 N(r) (t a e) E a e r ) = . " dt ( " 2r( r + 1 - (3) o cosh t - cos e

lim ar+ 1 -{3E+ (a, e , r) E e \ {O} , lim ( 1 - af+l-{3L (a, e , r) E e \ {O} , a-+O+ a--+ 1-

65

2) ASS?Lme (b- = ° and b+ =1= O) ar (b+ , b- E lR and 'Y > O). lf r is odd (respectively T is even) , then

E (a , e, r) =1= ° 'Va E ]0, 1 [ <=> e =1= 7r (respectively e =1= O ) .

3) Assume 'Y > O . lf r i s odd (respectively r is even) , then

� ° ( . ' l e \ , " ( e . \ J " W , ,, , r tf = respecnve y = 7r) =;> " a , , 7' ) T u v a E J U , 1 l .

Proaf. 1 ) For a > 0 , Izl ::; 1 , �a > 1 , we have

+00 zp 1 +00 ¡+oo 1 ¡+oo t<>- l e- (a- l ) t """ = -- ¿ zP t<>- l e- (a+p)t dt = -- dt � (a + p)<> r(a) p=o o r (a) o et - z '

It follows that

" ( e ) � e-ip() ( )r+l -il! � . eipl! " a, , r = c+ � (p + a)r+l-{3 + - 1 e c_ � (p + 1 _ a)r+ l-{3

with

1

.

¡+oo tr-{3e( l -a) t .¡+oo tr-{3e(a- l ) t

f( (3) ( c+ t _ '() dt + (- lf+lC 'I! t dt ) T + 1 - o e - e ' o e-' - e-

= ! " dt l ¡+oo tr-{3 N(r) (t a e) 2r(r + 1 - (3) o cosh t - cos e

N(r) (t , a, e) = c+ e-at ( et - eil! ) + (_ lf+lceat (eiO - e-t ) . The behaviour of the functions E± (a , e , r) is clear .

2) If b_ = 0, we get

N(r) (t , a, e) = b+ (e-at (et - cos e) + (- lf+l eat (cos e - e-t ) - i sin e (e-at + (- lf eat ) ) . Then , for b+ =1= 0 , a direct computation shows that if r is odd (respectively r is even)

1 1 E(a , e, r ) = ° <=> e = 7r and a = 2 (respectively e = ° and a = 2 ) '

66 F. BASTIN, C. BOIGELOT AND P. LAU B IN

If b+ , L E IR, we separate real and imaginary parts and get

N(1' ) ( t , a , B) = c+e-at (et - cos B) + c_ (- Ir eat (e-t - cos B) - i sin e (c+e-at + c_ (_ I ) " eat )

c+e (-a+ 1 ) t + ( - lrc_e(a- l) t - cos e (c+e-at + ( � lrc_eat) -i sin e(c+e-at + (- Ir c_ eat ) .

Assume now c± > O. For r even, we get directly that if &(a, e, r ) = ° for sorne a E ]0 , 1 [ , then 8 = O. The converse also holds since

lim &(a, 0, r) = +00, lim &(a, 0, r) = - oo . a-+O+ a-+l-

Let us assume now that r is odd. If a E ] 0 , I [ is such that &(a, e , r ) = 0 , a look at the real part of N(r) ( t , a, e) shows immediately that e =f O ; moreo'(er, since

we have

° = sin e �& (a , e , r ) + (1 - cos e) CS&(a , e , r )

= . tr-f3 + dt sin e ¡+OO.

C e-at (et - 1 ) + c_ eat ( 1 - e-t) 2f(r + 1 - {3) o cosh t - cos e .

This im plies sin e = ° and finally e = 7f . As in the previous case , the converse also holds .

3) For r even and e = 7f (respectively r odd and e = O) , we get

Taking real parts proves the assertion., O

Remarks. 1 ) If we consider

Q!u(x) = ¿ I k l f3uke2ik1rx , k'¡'O

j

Q�u(x) = ¿: sgn(k) l k lf3uke2ik1rX k'¡'O

and, for all r E IR, Ko is compact from H;er into H;;! and if the operator u � b+Q!u + LQ� u + uo satisfies the inequality of Theorem 7, then there is a compact operator Kl ; H;tf3 (]O , I D -+ H;:!(]O, I D such that

sup I (Af , g)Hm I � c l l f l lHs+p - I I Kd l lw+p (2m) per per per 9EVj , l Ig I l H2m-s =1 ' per

Rev. Un . . M ato A rgentina; Vol. 44- 1


for every j and every f E vtt1) . If A is bij ective from H;t! into H;er > a classical compactness argument shows that this inequality remains also valid with K1 = O and a smaller constant e.

2) For s > 1/2 , the norm 1 1 . lls related to the scalar product

(u, v ) s = u(O)v(O) + L I k l 2sUkVk I k l >o

is equivalent to the norm 1 1 . I I H�e, ' Under the assumptions of Theorem 7, if A is bij ective and s + (3 > 1/2 , 2m - s > 1/2 , we obtain that there is e > ° such that

sup I (Af , g)Hm I � e l l f I I Hs+il , f E V(r+ 1 ) ( 2m. ) pcr per J ,8

9EVj , I l g l l H2m- s =1 pe,

if and only if there is el > O such that

sup I (Af , g)m I � e/ l l f l l s+f1 ' f E vtt) · gEvym) , l l g l ! 2m_s =1

This is essentially due to the fact that the difference between the two scalar products involves compact operators . The proof is straightforward and uses classical techniques .

3 . 3 N umerical approach

3.3.1 Int roduction

If O is a smooth bounded and connected open subset of JR2 whose boundary ao is also connected, the norm 1 1 . l l m defines the usual topology on Hm (aO) .

Let O E [O , 1 [, T, m E N and

A = b+Q� + b_Q� + Ko

be a bijective pseudodifferential operator as introduced before and such that the operator u I-t b+Q�u + b_Q�u + Uo satisfies the conditions of Theorem 7. To

obtain an approximate solution of Au = f, one looks for Uj E VYó+1 ) such that the collocation equations

are satisfied. These equations are equivalent to

( A ) ( f ) E V(2m) Uj , v m = , v m ' V J •

( 1 )

(2)

Indeed , both (2 ) and ( 1 ) define a 2j x 2j linear system and 2m integrations by parts show that ( 2 ) is a consequence of ( 1 ) . Moreover , for large j , it follows from Céa's lemma (see appendix) that the system (2 ) has full rank since Theorem 7 gives the strong ellipticity of the operators in the used spaces.


68 E BASTIN, C . BOIGELOTAND P. LAUBIN

Using Theorem 7 and the fact that we deal with Riesz bases, we obtain that the l2-condition numbers of the Galerkin matrices are bounded uniformly in j (Proposition 9) . Hence , using the equivalence between the Galerkin and the collocation systems, the cürtdition numbers of the collocation matrices have also a good asymptotic behaviour after preconditionning. Without this manipulation , we do not obtain a good condition number for the collocation matrices , but we do for the double layer potential (see Example 10) .

3.3.2 Theoretical results on the l2-condition numb er

Let us introduce sorne notations for the next result (Proposition 9) . For every integer j � O, we consider a Riesz basis {u�s+f3) : p E Z, p � O} of H;�! (l O , I D (with bounds b , B) satisfying the following property : the functions u�s+f3) (p = O , . . . , 2j - 1 ) form a basis of V;�+ l ) . In the same way, we consider a Riesz basis {v�s+f3) - : p E Z, p � O} of H;��-S (] O , I D (with bounds e, C) of splines of order 2m satisfying similar conditions. Wavelets bases fulfil these conditions .

P rop osition 9 Assume that � < s < 2m - � , � < s + f3 < r + � . Jf

is bijeetive and is sueh that the operator u t-+ b+Q!u + b_Q�u + Uo satisfies the eonditions of Theorem7, then there is R > O sueh that the matrices Aj of dimension 2j defined by

satisfy A ( (A (s+f3) (2m-s» ) ) j = uq , vp m O�p,q<2i

T}2 (Aj ) = I I Aj l l 2 1 1Aj I I 1 2 � R�� if j is large enough. The eonstant R depends on the norm of the operator A (hence on the boundary of the domain) and on the constant of coercivity e (Theorem 7) .

Proof. The conditions on s , r, m , f3 come from the fact they we use Theorem 7 and Riesz bases . The proof goes in the classical way using Riesz bases in the right spaces and Theorem 7. O

For the collocation case , we also introduce sorne notations . If Bj = {u� , l = O, . . . , 2j - 1 } is a basis of VJ�+1) , we study the behaviour if

j -7 +00 of the condition'numbers of the collocation matrices A:i defined as follows

where

SPLINE WAVELETS IN PERIODfC SOBOLEV SPACES 69

We shall now discuss sorne exarnples that will be irnplernented nurnerically in Section 3 . 4 . below. We only want to treat sorne exarnples , used in our nurnerical exarnples. First , we consider the following bases

where

Ej = { l } U {Wo,r+ l ;i ,k , O ::; i < j , O ::; k < 2i }

+00 u� (x ) = 2j/2 L N8,r+I (2

j(x - l) ) , No,7'+ I (x) = Nr+I (x - 15) ,

1=-00 and where the functions Wo,r+I ;i ,k are constructed as usual starting frorn Wo,r+ I , which is the l-periodization of Wr+I (x - 15/2) . We shall also consider r = 2m - 1 and the bases

E" . = { l } U { 2i (m-s) e . k 0 < i < J' O < k < 2i } . J m,'t , - , -In this case, because the order of the splines is even , we choose 15 = O.

Example 1 0 Assume that � < s < 2m - �, � < s + f3 < r + � and that the operator

u I-t b+Q�u + b_Q�u + (u) o satisfies one of the conditions 1), 2) of Theorem 7. Then there are constants r l , r2 > O such that

} < r 2j l,6 l+s+,6 _ 2

Proof. First we consider the case

where (u)o is the zeroth Fourier coefficient of u. For u E Vt8+1 ) , sorne cornputations give

2j - 1 (Au) (kT

j) = (u) o + L e-2i7rÓ2-j l e2i7r2-j kl l l l r+ Idl (U) l

1=1 with (U) l = Jol dx e-ix2-rr1u(x + 2-j15) (the Fourier coefficients of u are (u) l e-2i1fól2-j ) andl

+00 j ,6 d - "\"' -2i1rpó (b b ( l 2j ) )

I l + p2 I 1 - D e + + - sgn + p (l + p2j )r+I . p=-oo l the complex numbers di are the same as in Theorem 7 .


The matrix S defined by

(S)lk = 2-j/2e-2i1f2-j lk , l , k = O , . . . , 2j - 1

is unitar� and diagonalizes the collocation matrix A:j . From this , we find that the eigenvalues Ak (k = O, . . . 2j - 1) of A7 are

and

2j (uJ ) o , 2je-2i1fÓ2-j k (uJhdk l k I T+1 k = 1 , . . . 2j - 1

(ASj ) = SUP{ I Ak l : k = O, . . . 2j - 1 } r¡2 J inf{ I Ak l : k = 0, . . . 2j - 1 } ·

Using the result on the behaviour of dk and (uJh we finally obtain the annouced result o

Now, we simply consider change of bases in V;�+l) . If ej denotes the matrix used to change the bases , we get

As} - e .ASj j - J j with r¡2 (ej ) bounded uniformly for j E N. In case r = 2m - 1 , the matrix change of bases B'· = B9 to B�+f3 has a condition number equals to 2(!+f3) (j- l ) . the matrix J J J ' change of bases B;+f3 to B; has a bounded condition number. Hence the conclusion .

For the general expression of the operator , we use the previous result and the one on the condition number of the Galerkin matrices (Proposition 9) . Let us denote by Kj the collocation matrices constructed using the compact operator Ko . We use the notation Aj for any of the collocation matrices constructed in the first part of this example ( i . e . with the particular expression of the operator ) ; finally, Gj (resp . Gj ) denote the Galerkin matrices corresponding to the particular expression of the operator (resp. · for the general expression) .

Because collocation and Galerkin systems are equivalent (for j large enough ) , there are inverse matrices Lj , independent of the chosen basis for the collocation matrices , such that

From the first relations ( 1 ) and the fact that the Galerkin matrices Gj have a bounded condition number we obtain that the behaviour of the condition numbers of the matrices Lj and Aj are the same. Now, the second relations (2 ) and the fact that the Galerkin matrices Gj have a bounded condition number show finally that the behaviour of the condition numbers of the . matrices Aj and Aj + Kj are the same. O

3 . 4 Sorne numerical exarnples

We have performed sorne numerical experiments to test the asymptotic convergence and stability obtained using the 8m;j,k functions . We used these functions because


S PLINE WAVELETS IN PERIODIC SOBOLEV S PACES 7 1

they have a ver y short support ; this fact , and also the fact that bases of wavelets are hierachical ones , lead to computations that are very easy to handle . We do not try to use the dual space because the functions are obtained from functions that are not compactly supported , hence the exact computations are rather heavy. The collocation matrices are preconditioned to get the Galerkin matrices añd then a bounded condition number .

We consider the double layer potential ((3 = 0 , m � 1 ) and the single layer potential ((3 = - 1 , m � 2) . For the experiments, we use the same basis for the test and trial bases although lower order splines can be used as test functions without great loss of accuracy. We always find a bounded sequence 1}(Aj ) and optimal order of convergence.

The computation of the elements of the matrices Aj can be performed easily using the Gauss-Legendre method with weights. It allows us to deal with a logarithm singularity in the case of the single layer potential .

We choose the connected open subset of JR2 represented on Figure 1 .

2 , 5 5 7 .

- 7 .

Figure 1

Its boundary is smooth, connected and given by

,(t) = ( (cos (6nt) + 8) cos (2nt ) , (sin(4nt) + 8) sin(2nt)) , t E [0 , 1] .

To use as efficiently as possible the almost orthogonality of the functions 8m,jk defined on [0 , 1 ] , we work with a parameterization by arc length. This requires to solve an autonomous differential equation . This can be done using a standard integration method before the computation of the matrices and takes a very short time since it depends linearly on the number of points.

Figure 2 gives the C2-condition number 1} of the matrix Aj for the double and the single layer potential on ,. This figure also gives the L2 norm of the error for the right-hand si de f (x) = 2 sin (2nx) and the estimated exponent of convergence (eoc) .

For the double layer potential, we use the Riesz basis { 1 , 8m;j,k : j � 0 , ° ::; k < 2i} and we treat the cases m = 1 (linear splines ) , and m = 2 (cubic splines ) .

For the single layer potential , we use the Riesz basis { 1 , 2 � 8m;j,k : j � 0 , ° ::; k < 2j } . Since the trial and test functions are chosen to be the same , we cannot use

. he re the linear splines 81 ;j,k because they do not satisfy the theoretical conditions of Proposition 9.

Double layer Simple layer potential potential

m = 1 I m = 2 m = 2

r¡ error eoc I r¡ I error eoc r¡ I error I eoc 1 5 .99 7 . 1 e-0 1 2 . 1 5 2 . 6 e-01 6 . 04 4 . 4 e-02 2 6 .90 2 . 0 e-0 1 1 . 82 33 .73 9 .0 e-02 1 . 55 24 .68 4 . 1 e-02 0 . 1 0 3 7 . 16 5 .2 e-02 1 . 96 50 . 29 6 .0 e-03 3 . 9 1 73 .57 6 .8 e-03 2 . 58 4 7 .23 1 . 3 e-02 1 . 95 55 . 32 5 .4 e-04 3 .47 8 1. 7 1 7 . 1 e-04 3 .27 5 7 .25 3 .3 e-03 2 . 0 1 55 .62 1 . 9 e-05 4 .83 82 .66 2 .0 e-05 5 . 1 6 6 7 .25 8 . 0 e-04 2 . 06 55 .64 8 . 3 e-07 4 .52 82 .89 7 . 6 e-07 4 . 70 7 7 .25 1 . 7 e-04 2 . 27 55 .64 4 .0 e-08 4 .37 82 .95 4 . 3 e-08 4 . 13

Figure . 2

3 . 5 Appendix

Céa's lemma Let X, Y be Banach spaces, let A E L(X,Y) be bijective and Jet Vj e X (j E N) ,

Tj e y' (j E N) , be sequences of subspaces such that dim(Vj) = dim(Tj ) < +00 for every j . Assume that (i) there are Pj E L(Y' , Tj ) (j E N) such that lirnj_>+CXl Pj (f) = f in y' for every f E Y' , (ii) there are b > O and a compact operator K E L(X, X) such that , for every j and u E Vj :

sup I v (Au) 1 2: b l lu l l x - I IKu l l x . vETj , I Iv l l yl = 1 Then there is No > O such that, for every j 2: No and u E X , the equation

has a unique soJution Uj E Vj . Moreover, there is C > O such that

I l u - uj l l x ::; C inf I l u - w l l x . wEVj

Here we use this lernrna with

References

V. - V(r+l ) J - j ,8 , T - { . V(2m) } j - . < . , 9 > H;J;,r(1R) · 9 E j '

[1 ] N. Arnold and W.L. Wendland, On the asymptotic convergence of collocation

methods . Math. Comp. 4 1 , 1983 , 349-38 1 .


[2 ] D .N . Arnold and W.L . Wendland, The convergence of spline collocation for strongly elliptic equations on curves , Numer. Matb. 47, 1 985, 3 1 7-341 .

[3] F. Bastin and P. Laubin, Regular compactly supported wavelets in Sobolev spaces , Duke Matb. J. 87, 1997, 481-508.

[4] F. Bastin and P. Laubin, Compactly supported wavelets in Sobolev spaces of integer order , Applied and Computational Harmonic Analysis Vol . 4, Number 1 , 1 997, 51-57 .

[5] G. Beylkin , R . Coifman, V. Rokhlin, Fast avelet transforms and numerical algorithms 1, Comm. Pure and Appl. Matb. 44, 1 991 , 141 - 183 .

[6] C.K . Chui and J .Z . Wang, On compactly supported spline wavelets and a duality principIe, Transactions of the AMS 330, 2 , 1992 , 903-9 15 .

[7] C. K . Chui, An introduction to wavelets , Wavelet analysis and its applications , Volume 1 , Acad. Press 1992.

[8] M. Costabel and E . Stephan , On the convergence of collocation methods for boundary integral equations on polygons , Math. Comp. 49, 1 987, 46 1-478.

[9] S. Dahlke , W. Dahmen, R. Hochmuth, R. Schneider , Stable Multiscale Bases and Local Error Estimation for Elliptic Problems , Appl. Numer. Matb. 23 ( 1 ) , 1997, 2 1-48.

[10] W. Dahmen, Stability of multiscale transformations, The Journal of Fourier Analysis and applications Vol. 2, Number 4, 1996, 341-361 .

[ 1 1 ] W . Dahmen, S . Prossdorf, R . Schneider , Wavelet approximation methods for pseudodifferential equations 1 : Stability and convergence ,Mathematische Zei tschrift 2 1 5 ( 1994 ) , 583-620.

[12] J. Elschner and LC. Graham, An optimal order collocation method for first kind boundary integral equations on polygons , Numer. Math . 70, 1 995, 1-3 1 .

[ 13 ] LG . Graham, C .A . Chandler , High order methods for linear functionals of solutions of second order integral equations , SIAM J. Numer. Anal. 25 , 1988 , 1 1 18- 1 137 .

[14] P . Laubin , High order convergence for collocation of second kind boundary integral equations on polygons, Numer . Math. 79 , 1998, 107- 140.

[ 15] P. Laubin and M. Baiwir , Spline collocation for a boundary integral equation on polygons with cuts , SIAM J. Num. Anal. 35 (4) , 1 998 , 1452- 1472 .

[ 16] Y. Meyer, R. Coifman, Ondelettes et Opérateurs I-II I , Hermann, 199 1 .

[ 17] S . Prossdorf and B . Silbermann , Numerical analysis for integral and reIated operator equations , Birkhauser , Operator Theory: Advances and Applications, 52, 199 1 .

Rev. Un. Mat. A rgentina. Vol. 44- /


[ 18] J. Saranen, The convergence of even degree spline collocation solution for potentia! problems in smooth domains of the plane, Numer. Math . 53, 1988, 499-512 .

[ 19] J . Saranen and W.L . Wendland, On the asymptotic convergence of collocation methods with spline functions of even degree, Math . Comp. 45, 1985 , 91 - 108 .

[20] W.L. Wendland, Boundary element methods for elliptic problems , Mathematical theory of finite and boundary elements methods , DMV Seminar , Band 15 , Birkhauser , 1990 .

Symbols

1 . Z is the set of integers

2. N is the set of strictly positive integers

3. Vj(m) (j E Z, m E N) is the set of functions on IR which are smoothest splines of degree m - 1 with respect to the mesh {2-ik : k E Z} and be long to L2 (IR)

4 . Vj�;') (8 E [0 , 1 [ , j E Z, m E N) is the same (as Vj(m) ) set of splines but with respect to the mesh { 2-i (k + 8) : k E Z}

5 . The corresponding sets of 1-periodic splines are respectively denoted by V;m) , V(m) i,6

6 . H�er (] O , 1 [) (8 E IR) is the Sobolev space of order 8 of 1-periodic distributions on IR

7. 'l/Jm

is the dassical Chui-Wang spline biwavelet ('l/Jm

E V1(m) )

8. ()m is the spline function in V1(2m) such that Dm()m = 'l/Jm and sUPP(()m

) e [0, 2m - 1]

Acknowledgment The. authors thank the referee for his comments and suggestions .

Address F. Bastin University of Liege , Institute of Mathematics (B37) , B-4000 Liege , BELGIUM Fax: xx32 (0)4 366 95 47; email : F . Bastin@ulg .ac .be

Note : P. Laubin died in 2001

Recibido : 1 9 de Junio de 2002. Aceptado : 8 de Febrero de 2003 .


REVISTA DE LA

UNJON M ATEMATICA ARGENTINA

Volumen 44, Número 1 , 2003, Páginas 75-88

FINITELY GENERATED MULTIRESOLUTION ANALYSIS IN SEVERAL VARIABLES

ANA BENAVENTEt AND CARLOS A. CABRELLlt

t INSTITUTO DE MATEMÁTICA APLICADA , UNSL AND CONICET . :j: FCEyN , UNIVERSIDAD DE BUENOS AIRES AND CONICET .

ABSTRACT . Let r be a lattice in ]Rn and A a dilation matrix such that Ar e r . Let 'P be a localized square integrable vector function and assume that the lattice translates of 'P are orthonormal. We give necessary and sufficient conditions on 'P in order that it generates a Multiresolution Analysis in ]Rn with respect to the

. lattice r and the dilation A. This characterization extends previous results to the case of regular non-compactly supported functions.

1 . INTRODUCTION

75

The concept of Multiresolution Analysis (MRA) due to Mallat [Ma189] and Meyer [Mey92] provided the first systematic way to construct orthonormal wavelet bases of .c2 (lR) . The structure of a MRA is generated by a function (the scaling junction) that satisfies a certain self-similarity condition. The problem of constructing orthonormal wavelets was then shifted to the problem of constructing MRA's . The theory was extended to several variables . To take full advantage of the higher dimensionality it is important to consider arbitrary dilation matrices (not only dyadic dilations) . This has proved to be useful in applications to image representation where the geometry of the picture is better described with matrices that adapt better to the situation . The side effect is that the theory becomes much more complicated and the results are not a straightforward generalization of the 1-dimensional case. Another important 'generalization is the case in which a finite number of generators for the MRA are allowed [Alp93] [GLT93] [GHM94] [CH96] [HSS96] [CDP97] [Ald97] [JRZ99] [Ca199] [CHM99] . This is known in the literature as MRA with multiplicity, and the associated wavelets as Multiwavelets . The framework of multiple generators provides much more flexibility to construct bases with predetermined properties . The characterization of MRA in this generality was done in [CHM99] for compactly supported functions . In the present article we work in the following context : let r be an arbitrary lattice in lRn , and A a dilation matrix compatible with the lattice r, ( i . e . A(r) e r and every eigenvalue ,x, of A satisfies l ,x, 1 > 1 ) . Let <P = (<Pl , . . . , <Pr ) , <P E .c2 (lRn , e) and <Pi belongs to the Sobolev space 1lm(Rn) , 'l/m E N. Assume that the lattice translates of <P are orthonormal . We give necessary and sufficient conditions on

1991 Mathematics Subject Classification. Primary :42C40 ; Secondary:42C30. Key words and phrases. Multiresolution Analysis, Wavelets, Multiwavelets, refinable equation. The research of the authors is partially supported by Grants: UBACyT TW84, and CONICET,

PIP456j98 and BID-802jOC-AR. PICT-03134.

76 A. BENAVENTE AND C. A . CABRELLI

the vector function <p in order that it generates a Multiresolution Analysis of ne . (Theorem 3 . 1 ) . These conditions were obtained by Albert Cohen for the l-dimensional case, scalar functions (r = 1 ) and later extended to the multidimensional setting for the case that the dilation matrix is 21 [Coh90] . In [CHM99] , Cohen's theorem was extended to include the case of arbitrary dilations matrices and a finite number of generators with compact support. The contribution of this paper is to show that these conditions can be extended to a much wider class of generators. We were able to proof that the hypothesis of the generators to be compactly supported can be relaxed . We assume instead certain decay of <p. More precisely we require that for i = 1, . . . , r, each <Pi belongs to the Sobolev space 1-lm(Rn) , '7m E N. The proof, as the ones in [Coh90] and [CHM99] , i s a "time-domain" proof in the sense that it doesn 't use the Fourier Transform. The main argument is based on a counting technique related to the geometric properties of the tiling associated with the dilation matrix. In the case in which the dilation matrix is A = 21, the tile element is a cube; then the geometry is simple and the integrals that have to be estimated are integrals over cubes in ]Rn . When one allows arbitrary dilation matrices , the associated tile can be of a very complicated geometry and also have fractal boundary. This makes the estimation of the integrals much more involved , and the counting results are more complicated to obtain. The removal of the assumption of compact support for the scaling vector , requires a refinement of the techniques in order that the counting results can be applied to this more general case. Necessary and sufficient conditions for when a nested sequence of 2k-dilated principal shift invariant space (PSI) , has dense union and zero intersection where obtained in [BDR93] for the one dimensional case. A PSI is a shift invariant space generated by a single function . The generator in this case doesn't need to be an orthonormal basis neither a Riesz basis of the closure of the span of its integer translates . This general condition is expresed in terms of the zeroes of the Fourier transform of the generator . This setting differs from Cohen approach in the sense that Cohen's Theorem characterizes exactly orthonormal MRA's . The PSI here is generated by ,a function that has orthonormal translates . Later , Jia and Shen [JS94 ] formulated the conditions ort [BDR93] , (see also [Shen98] ) for the finitely generated case (FSI) , given some indication of the proof. The characterization in our paper is in the direction of the approach of Cohen. We characterize orthonormal MRA's for the case of a general dilation matrix, compatible with an arbitrary lattice, in higher dimensions , for several functions . The organization of the paper is as follows . In Section 2 , we briefly review the concepts of lattices , tiles , Multiresolution Analysis and the relation between them in terms of the generalized Haar's MRA. In Section 3 we state our main result in Theorem 3 . 1 . For a better organization of the proof, in the following subsections, we discuss and prove in Propositions 3 . 2 , 3 .4 and 3 . 5 the necessary and sufficient conditions that have to be satisfied by the localized vector scaling function to generate a MRA. Finally, in subsection 3 .4 we combine the results of the Propositions to prove Theorem 3 . 1 .

FINETELY GENERATED MULTIRESOLUTION ANALYSIS 77

2. LATTICES , TILES AND MULTIRESOLUTION ANALYSIS

Let r be an arbitrary lattice in ]Rn ( i . e . , r = R(zn) with R any invertible n x n matrix with real entries ) . Let now 1C be a fundamental domain for this lattice e .g . , 1C = R( [O, I )n ) and set K, = Idet(R) I . Let A be a dilation matrix for r, i .e . , A(r) e r and every eigenvalue A of A satisfies IA I > 1 . The determinant of a dilation matrix for a lattice is always an integer and its absolute value is the number of cosets of the quotient group r / A(r ) . A digit set for A and r is any set of representatives of this group . Let q = I det (A) I . We assume that there exist a digit set D === {do , . . . , dq-d for A and r such that the set Q === n:=:l A-k�k : �k E D} has n-dimensional Lebesgue measure K,. Without Joss of generality we will assume that do = O. For a general dilation matrix it is not always true that such digit set exists . (A counterexample was found in [Pot97] ) . If this set of digits exists , we will say that A is an admissible dilation matrix. The set Q is compact and tiles the space by r -translates in the sense that the r-translates {Q + khEf cover ]Rn with overlaps of measure zero . Moreover, they satisfy the following self-similar condition (See [GM92] , [Hut8 1] ) :

q- l A(Q) = U Q + d8 •

8=0 Given a function g : ]Rn -7 Cr , A a dilation matrix, q = I det A l , j E Z and k E r, we will write gj,k (x) = qj/2g(Ajx - k) , to denote a translation of g by A -j k followed by an .c 2 -normalized dilation by Aj .

2 . 1 . MULTIRESOLUTION ANALYSIS . A Multiresolution Analysis (MRA) of multiplicity r associated to a dilation matrix A and a lattice r is a sequence of closed subspaces {Vj }jEZ of .c2 (]Rn) which satisfy: P I Vj e Vj+l for each j E Z, P2 g(x) E Vj � g (Ax ) E Vj+l for each j E Z, P3 n Vj = {O} , jEZ P4 U Vj is dense in .c2 (]Rn) , and jEZ P5 there exist functions 'Pl , " " 'Pr E .c2 (]Rn) such that the collection of lattice

translates {'Pi (X - k)hEf, i=l , ... ,r forms an orthonormal basis for Va . If these conditions are satisfied, then the vector function 'P = ('Pl , . . . , 'Pr? is referred to as a scaling vector for the MRA. Gréichenig and Madych [GM92] established a connection between self-similar tilings and multiresolution analysis that have a characteristic function for a scaling function. They showed that there is a Haar-like multiresolution analysis associated to each choice of dilation matrix A and a digit set D for which the set Q is a tile . In particular , they proved that if Q is a tile, then the scalar-valued function X Q generates a multiresolution analysis of .c2 (]Rn ) of multiplicity 1 . Note that the fact that {X Q (x - k) hEf forms an orthonormal basis for Va is a restatement of the assumption that the lattice translates of the tile Q have overlaps of measure zero . An immediate consequence of Gréichenig and Madych's generalization of the Haar's multiresolution analysis is the following:

Re\'. Un. Mat. A rgentina. Vol. 44- 1

78 A . BENAVENTE AND C . A . CABRELLI

Lemma 2 . 1 . The collectíon

{X{t}iEz.kEr = {qi/2XQ(Aix - k) }iEz.kEr is complete in .c2 (lRn) i . e . , its finite linear span is dense in .c2 (lRn) . 2 .2 . LOCALIZED VECTOR SCALING FUNCTIONS . We say that a MRA in .c2 (lRn) is localized or regular if the scaling vector ep = (epi , . . . , epr f is localized in the sense that for i = 1 , . , . , r, each <Pi belongs to the Sobolev space 1lmCJ?;1-), Vm E N. This condition is equivalent to: for each i = 1 , . . . , r,

(2 . 1 )

Using Cauchy-Schwarz 's inequality and taking into account that the function ( l+ l�l )m belongs to .c 1 (lRn) with m > 1 , it is easy to prove that if a function I satisfies ¡ E 1lm(Rn) for all m E N, then:

(2 . 2) { I l (xWdx ::; ( 1 c

�)m ' JI I:z: I I�M +

(2 .3)

(2 .4) , ,

For simplicity, we shall from now on write that the vector function ep has orthonormal lattice translates when we mean to say that {epi (X - k) her.i= l , . . . ,r i s an orthonormal system in .c2 (lRn ) . Definition 2 .2 . Assume that ep E .c2 (lRn , ccr) has orthonormal lattice translates . Let Vo be the closed linear subspace generated by the lattice translates of epi , i . e . ,

(2 .5 ) Vo = span{epi (x - k)hEr,i=l , . . . . r . For each j E Z let Vi be the set of all the dilations of Vo by Ai , i . e . ,

. ' k (2 .6) Vi = {g (AJx) : 9 E Vo} = span{epi ' : i = 1 , . . . , rhEr .

If {Vi};a is a MRA for .c2 (lR") , then we say that the MRA is generated by ep. Remark 2 .3 . In the characterization of MRA due to A. Cohen [Coh90] , he uses a localized generator , but the dilation matrix is the uniform one : A = 21. In the proof, he uses the essential fact . that 2I maps dyadic cubes into dyadic cubes . This is not possible in the case of arbitrary dilation matrix.

3 . NECESSARY AND SUFFICIENT CONDITIONS .

We now are ready to state the main result in the paper :

Theorem 3. 1 . Let ep = (epI , " · , eprf E L2 (Rn , cr) such that lor each i = 1 , ' " , r, <Pi E 1lm(Rn) for all m E N and that the set {epi ( ' - k) hEr. i= l . . . . ,r is an orthonormal system. Let A be an admissible dilation matrix for the lattice r. Then ep generates a multiresolution analysis with multiplicity r associated to (r , A) if and only if: Rev. Un. Mat. A rgentina, Vol. 44- 1

FINETELY GENERATED MULTIRESOLUTION ANALYSIS

a) 'P satisfies a refinement equation of the form

'P(x) = L ck'P(Ax - k) kEr

for some r x r matrices Ck = (c�j ) , su eh that for eaeh i , j = 1 , . . . , r, { ct hEZn , E e2 (r ) and

r b) 1 1<p(O) 1 1

2 = L l <Pi (O

W = I Q I · i=l

79

To prove this theorem, in the next propositions we will give necessary and sufficient conditions on the vector function 'P, in order that the subspaces V j will satisfy properties P I , P3 and P4 of the definition of MRA. Property P2 is satisfied from the definition of Vj and P5 is assumed .

3 . 1 . PROPERTY ( P I ) : Vj e Vj+1 ' Proposition 3 .2 . Let 'P = ('PI , . . . , 'Prf E .c2 (IRn , cr) with orthonormal lattice translates. Let A be a dilation matrix and define Vj as in (2. 5) and (2. 6) . Then, the following conditions are equivalent:

(1) Vj e Vj+l for all j E Z. (2) The vector funetion 'P i s refinable, z. e . i t satisfies the refinement equation :

'P(x) = L Ck'P(Ax - k) kEr

for some r x r- matriees Ck , sueh that for eaeh i , j = 1 , . . . , r, the sequenee of coefficients {c7,j hEr is in e2 (r) .

Proof: If ( 1 ) is satisfied, then 'Pi E Vo e VI for i = 1 , ' " , r. The definition of the subspaces Vj implies that {ql/2'Pj (Ax - k) }kEr ,i ,j= I , . . . ,r = {'PY (x) }kEr is an orthonormal basis for VI , then the representation of each <Pi respect to the orthonormal basis of VI will be:

r (3 . 1 ) 'Pi = L L c7,j'PY (in .c2 (IRn) ) ,

j=1 kEr where C�j := < 'Pi , 'PY > . For i , j = 1 , . . . , r, the sequence of coefficients {C�,JkEr belongs to e2 (r) . Let us call Ck the r x r-matrix whose columns are (C�I ' . . . , c� ) . Considering that 'P = ('PI , . . . , 'Pr )T then, from (3 . 1 ) we have 'P = ¿kEr Ck'P1 ,k in .c2 (lRn , cr ) , or equivalently

kEr and condition (2 ) is satisfied. For the converse, if 'P is refinable, then 'Pi E VI , i = 1, . . . , r so Vo is included in VI .

O

3 . 2 . PROPERTY ( P3) : njEz Vj = {a} . We shall prove that (P3) is a consequence of the orthonormal r-translates of 'P and the localization of each 'Pi . To do this , we will use the following lemma (we omit the proof because it is like in the classical l-dimensional case with dyadic dilations [Woj97] ) :

Rev. Un. Mat. A rgentina, Vo!. 44- 1

80 A. BENAVENTE AND C. A. CABRELLI

Lemina 3.3 . Consider Vj as in (2. 6) . Let Pj be the orthogonal projectiori of .c2 (lRn) onto Vj . Suppose that for all 9 E .c2 (lRn) , , lim I IPjg l 1 2 = O . J -4 - oo Then n Vj = {O} .

JEZ Proposition 3.4. Let ep E .c2 (lRn, Cr) be a localized vector function in the sense of (2. 1) . Suppose that ep has orthonormal lattice translates, A is a dilation matrix and consider the subspaces V j as defined in (2. 6) . Then n V j = {O} .

JEZ Proof: Using Lemma 3 .3 , it suffices to show that limj-4-oo I I Pjg l 1 2 = O , '<:/g E .c2 . Moreover, it suffices to establish this limit for 9 contained in a subset whose finite linear span is dense in .c2 (lRn) . We will use the complete set gíven in Lemma 2 . 1 , i .e . we will prove that

(3 .2 )

Fix any s E Z and e E f . Since q = I det(A) I , we have for every j E Z that

I Aj-S ( Q + e) 1 = qj-s l Q + e l = qj-s I Q I . Since {cp{,khEr,i=l , . . . ,r is an orthonormalbasis for the subspace Vj then,

I I PjX�e l l � = j�S t z= I r . epi (X - k)dx l 2 q i=l kEr J AJ- S (QH) Using Cauchy-Schwarz's inequality, we therefore compute that

I l PjX�e l l � :::; IAj-S��s + e)1 t z= 1 . lepi (X - kWdx q i=l kEr J AJ-' (QH)

(3 .3) = I Q I tz= l l ep; (xWdx, i=1 kEr J AJ- s (Q+e)-k

where the last sum is finite . To see this, note that for a fixed s and j < s , using that Q is a tile for lRn and A - 1 is contractive, we have that the lattice translates of Aj - s Q have overlaps of measure zero . To simplify the notation, write E : = A-S ( Q + e); then Aj-s ( Q + e) - k = Aj E - k . Choose an integer J < O small enough such that I Aj E - k n Aj E - k' l = O for all j :::; J; k, k' E f, k ¡:. k' . Then, for j :::; J,

z= XAjE-k (X) lepi (XW = XUkEr AjE-k (X) l epi (X) 1 2 kEr

And since l epi (X) 1 2 E .c1 (lRn) , then

z= r l epi (XWdx = r Z= XAjE_k (X) I c¡Ji (X) 1 2dx < oo . kEr J Aj E-k J'Rn kEr

Now, using ( 3 . 2 ) and (3 .3 ) , it suffices to prove that for i = 1 , . . . , r ¿kEr JAj E_k l epi (X)1 2 dx Rev. Un. Mat. A rgentina, Vol. 44- 1

FINETELY GENERATED MULTIRESOLUTION ANALYSIS 8 1

goes to O when j -+ - oo . Consider the same integer J E Z as before and j ::; J; and define fj (x) : = L XAjE-k (X) \ IPi (XW · So fj (x) = \ IPi (X) \ 2XukHAjE-k (X) , Then:

kEZn

L { \ IPi (X) \ 2dx = { Ji(x)dx kEf J Aj E-k JRn

= ( fj (x)dx J[-p,pln

= h + h

Write Bj : = U (AjE - k) n [-p, pt, then kEr

(3.4)

We can see that \ Bj \ --7 O as j -+ - oo . In fact: consider tj : = card ( {k E zn : Aj E - k n [_p, p]n i- 0} ) and write ó ( · ) as the diameter of a certain set o Because the spectra1 radius p of A-I is 1ess than 1 , we have that I IAj \ \oo -+ O when j -+ - 00 (see [HJ] ) i .e . \ \Aj \ \ oo < E for j small enough. Then, considering the metric d(x, y) = \ \ x - y \ \oo = max{ \ xi - Yi \ : i = 1 , . . . , n} we have that for x , y E AJE:

d(x , y) = \ \Aj (A1-jx - AI-jy) \ \00 ::; \ \Aj \ \00 \ \A 1-J X - AI-jy \ \oo < d(A1-jx , AI-jy) ::; ó (AE) .

Taking the supremum over Aj E, we have that ó(Aj E) ::; ó(AE) and then tj < tI , for all j < O . Now, for j ::; J, the overlaps of the 1attice trans1ates of AjE have me asure zero , so:

\Bj \ = tj I Aj E - k n [-p, pt l

::; tI IAj E l = tI IE \ qj < C .

Since \ Bj \ -+ O as j -+ O then fjX [-p,pln -+ O , moreover \ JiX[-p,pl n (X) \ ::; \ IPi (X) \ 2 .

From this fact , equa1ity (3 .4) and the Dorninated Convergence theorern we have that

(3 . 5) h = ( fAx)dx -+ O when j -+ - oo . J[-p

,pln

For the integral h , take e > O and j ::; J. Then O ::; Ji (x) ::; \ IPi (x ) 1 2 . U sing this and property (2 . 2) for sorne m, we have

12 ::; { \ IPi (xWdx J1 1x l l ?p

< Cm - ( l + p)m '

82 A . B ENAVENTE AND C. A . CAB RELLI

Considering p large enough such that ( l;;;)m < e, we will have 12 < e . Finally :

r fi (x)dx = Il + h ::; Il + e . JlRn Taking limit for j -+ - 00 , from (3 . 5) we conclude that for all s E Z and f E r, limj--+_oo I I PjX:{ 1 1 2 = O . Hence n Vj = {O} .

jeZ

3 .3 . PROPERTY (P4) : U Vj = .c2 (]Rn) . jEZ

o

Proposition 3.5 . Let cP = (CPl , . . . , CPr)T E .c2 (]Rn , cr) a localized vector function. Suppose that cP has orthonormal lattice translates . Let A be a dilation matrix and let Vo and Vj be defined as (2. 5) and (2. 6) respectively. If (3 .6) t l<Pi (OW = t i J CPi (x)dx I 2 = I Q I i= l i=l Then UjEz Vj is dense in .c2 (]Rn) . Reciprocally, if UjEz Vj is dense in .c2 (]Rn ) and cP is refinable, then (3. 6) is satisfied.

Before proving this proposition, we are going to present sorne auxiliary results with respect to the decomposition of ]Rn by the tiles { Q + k hEr . First , note that the fact that Q is self-similar together with the fact that the translates of Q tile ]Rn with overlaps of measure zero , implies that the dilated tile Aj Q, j 2:: 1 is a union of exactly qj translates of Q, with each of the translates lying entirely inside Aj Q. Following the idea in [CHM99] , for j 2:: 1 we are going to split the lattice r into a finite set containing those elements that translate Q entirely inside Aj Q, and a finite set containing the elements that translate Q to the boundary of Aj Q . More precisely, for each j 2:: 1 let us consider the following finite subsets of r :

(3 . 7)

Nj = {k E r : Q + k e Aj Q} , N? J {k E Nj : Q + k e (Aj Qt} ,

N? J {k E Nj : ( Q + k) n 8(Aj Q) -I 0} .

·These sets satisfy the following relations : Aj Q = Q+Nj , card(Nj ) = qj , NiUNa = Nj and Ni n Ny = 0 . Let n = {k E r : ( Q + k) n B -1 0} . The following technical lemma (see [CHM99] for a proof) characterizes those translates Q + 'Y of Q for which it is possible to translate Q + 'Y by elements of n so that one translate Q + 'Y + k with k E n lies entirely within Aj Q and another translate Q + 'Y + k' with k' E n lies entirely outside of Aj Q (neglecting its boundary) . This lemma also tells us that the ratio of the number of those translates Q + k that intersect the boundary of Aj Q to the total number lying insid!'l Aj Q converges to zero :

Lemma 3.6 . Let B, n, Ni > Ni and NJ defined as befo re, then: card(NJ) card(NJ)

a) lim . . = 1 and lim . = O. j--+oo qJ j--+oo qJ

FINETELY GENERATED MULTIRESOLUTION ANALYSIS

. eard(Nj\ ( (NJ - O) n Nj ) ) b) hm , = 1 . j-+oo qJ

83

e) Let "Y E r, Ji there exist k, k' E O su eh that Q + k + "Y e Aj Q and Q + k' + "Y e �n\ (Aj Q)O , then "Y E NJ - O = {t' - w : e E N/, w E O} .

PROOF OF PROPOSITION 3 . 5 : Suppose that L I tPi (0) 1 2 = I Q I · jEZ i=l

Note that if for all 9 E .c2 (�n) (3 .8 ) then Property (P4) i s satisfied . Further , i f <p i s refinable , then by Proposi"tion 3 .2 , Vj e Vj+1 and therefore (3 ,8) i s equivalent to Property (P4) . Now, by orthogonaHty, I I Pjg - g l l � = I l Pjg l l � - l l g l l � , then we can rewrite equation ( 3 . 8) as :

(3 , 9) \lg E .c2 (�n) , lim I l Pjg l l � = I l g l l � . J-+OO This express ion is true for all 9 E .c2 (l�n) if and only if is true in a dense subset of .c2 (�n) . So we will use the set of functions considered in Lemma 2 . 1 . Then,

(3 . 10) Il Pj (X Q) I I � = 1j t L I r , <Pi (X - k)dX l 2 q i=l kEf JAJ Q

On the other hand, let s E Z, e E r, and j 2: s . By a change of variable and taking into account that Ar e r, we have

(3 . 1 1 ) Comparing (3 . 10) and (3 . 1 1 ) , we conclude that (3 .9) i s equivalent t o the statement :

(3 . 1 2) lim I I Pj (X Q) I I � = I IX Q I I � = I Q I · J-+OO Since statements (3 .8) , (3 .9) and (3 . 12) are equivalent , we conclude that it suffices

r to prove that lim I l Pj (X Q) I I � = L I tPi (OW , or equivalently, to prove that for each J-+OO ' i = 1 , . . . , r :

(3 . 1 3)

i=l

lim � L I r <Pi (X - k)dX l 2 = I tPi (0) 1 2 . J-+OO qJ J Aj Q kEI' To do this , fix 'i , consider a constant Mi > O and define Ki = {x E �n : I l x l l ::; M;} . Using the property o f unconditional convergence of orthonormal bases , we will split the summation over r into three disjoint regions related to the subset Ki . The idea behind this is that the first regio n should contain only elements k of the lattice r such that Ki + k is sure to He in the interior of Aj Q, the second region should contain those k for which this translation will intersect the boundary of Aj Q, and the last region should be the complement of the firsf two . More precisely, let Bi be any open ball in �n which contains both Q and Ki , and define

O = {k E r : ( Q + k) n Bi -=1= 0} . Rev. Un. Mar, A I;¡;e11fino, Vol, 44- 1

84 A . . BENAVENTE AND C. A. CABRELLI

Note that n is finite and Ki e n + Q. For each j 2:: 1, define:

r1 ,j = N'j\( (N; - n) n Nj ) , r2 3· = NI! - n , . 3 ' r3,j = f\ (r1 ,j U r2 ,j ) ,

where the sets Nj , Njo and NJ are as in (3 .7) . Note that for each j, the sets f l,j , r2,j , r3 ,j partition r . Further , by Lernrna 3 . 6 a) and b) , we have:

(3 . 14) lirn card(�l ,j ) = 1 and lirn

card(�2 ,j ) = O . j-HXJ q3 j-4oo q3

Now define

Rsj = 1j 2: I f . 'Pi (X - k)dx I 2 , S = 1 , 2 , 3

q kEr . JA1 Q . 8 ,1

We will show that : �irn Rlj = l cí?i (OW , lirn R2j = O and �irn R3j = O . )-+00 3-400 3-+00

Let us begin with R2j .

R2j ::; 1j 2: ( f . l 'Pi (X _ k) l dX) 2

q kEr2,j J A1 Q ::; � 2: ( f l 'Pi (X) l dx) 2

q3 Jan kEr2,j . C card(r 2 ,j )

= ---;-'-"':::":" qj

. By (3 . 14) , the last terrn is arbitrarily srnall for j Jarge enough. Then R2j 4- O when j 4- oo . To analyze R3j , Jet us write ¡Pi (X) = X K¡ (X)'Pi (X) , Then 'Pi (X ) = ¡Pi (X )+X K¡ (X)'Pi (X) , Note that ¡Pi has cornpact support .

R3j ::; 1j 2: ( l l ¡Pi (X - k) l dx + l IXKf (Y _ k)'Pi (Y _ k) l dY) 2

q kEr3,j JA1 Q JA1 Q = A + B + C,

We will show that A = B = O and C 4- O as j 4- oo. Suppose that A =1 O , then JAjQ l ¡Pi (X ,.c.. 'Y) l dx =1 O for sorne 'Y E r3,j ' Then (Ki + 'Y) n Aj Q has to have positive Lebesgue rneasure . Since Ki e Bi e Q + n, then (Ki + 'Y) e (Q + n + 'Y) and ( Q + n + 'Y) n Aj Q will have positive rneasure. Because Aj Q is the exact union of


qj translates of Q that do not overlap , then the only translates of Q that interseets Aj Q in sets of positive measure, are the translates that are eompletely inside of Aj Q. Henee:

(3 . 15 ) Q + k + 1 e Aj Q for somek E n. Sinee o E n and Ny e Nj , then Ny e (Ny - n) n Nj . Henee Nj = Nj U Ny e f1 ,j u f2 ,j . Sinee 'Y E f3,j = f\ (f1 ,j U f2 ,j ) , then 1 � f2,j , so 1 � Nj . This implies that Q + 1 is not eontained in Aj Q. Then Q + 1 e ]Rn\(Aj Q)o . Consequently

( 3 . 16)

and sinee O E n, then from Lemma 3 .6 e) , applied to (3 . 1 5 ) and (3 . 16 ) we have 1 E Ny - n = f2 ,j , and this is a eontradietion . Then A = O . By a similar reason, B = O. To prove that e � O as j -+ 00 , we writeT3,j as the union of disjoint sets as follows :

(3 . 17 ) 00

f3 ' = U Dj ,J • • =1

where for eaeh j E Z, Dt := {k E f3,j : s ::; dist (Aj Q. - k , O) < s + 1 } . After the ehange of variable x = y - k , we have :

e = � L ( r l 'Pi (X) l dX) 2 qJ kEr3j J Aj Q-knKf

::; 1j f L ( r . l 'Pi (X) l dX) 2

q 1 . JAJ Q-.k . .= kEm

::; 1j f L (1 l 'Pi (x) l dX) 2

q .=1 kED� . IIxll;:O:,

For a purpose that will beeome clear later eonsider m > 11n . Using that <Pi E 1{m(]Rn) and property (2 .4) , then

(3 . 18 )

·with Cm a eonstant that depends on m . Let us find an upper bound for eard(D� ) . Let us note that if 'Y E Dt then s ::; dist (Aj Q - 1 , O ) < s + 1 . Sinee Aj Q - 1 is eompaet, then there exists x E 8(Aj Q - 1) where the distanee is attained . On the other hand Aj Q - 1 is the union of exaetly qj tiles that do not overlap , henee x E Q - t for sorne t. Then Q - t e Aj Q - 1 and Q - (t + 'Y) n 8(Aj Q) i- 0. Finally t + 1 E Ny and D� e b E f 3,j : :Jt E D� , such that 1 + t E Nf} . 1t follows that

(3 . 19 ) eard(D� ) ::; eard(D� ) . eard(Ny ) .

Moreover eard(D�) ::; c(s + 1 + d) , with d = diam(Q) . To see that , take 1 E D� then Q - 1 e B (O , s + 1 + d) (the open ball eentered at zero, and radius s + 1 + d) . Then D� e L := b E f : Q - 1 e B(O, s + 1 + d) } and eard(D� ) ::; eard(L) . Now,

86 A. B ENAVENTE AND C. A. CABRELLI

since the lattice translates of Q do not overlap, we have :

(3 . 20) card(L) I Q I = 'L I Q - , I = U ( Q - ,) ""(eL

On the other hand

(3 .2 1 ) U ( Q - ,) :::; I B (O , s + l + d) 1 = c(s + l + dt· ""(eL

From (3 .20) and (3 .2 1 ) we have card(L) :::; c(s + 1 + d)n , and finally card(D� ) :::; c(s + 1 + d)n . By (3 . 19) it follows that card(D� ) :::; card(Nf ) . c(s + 1 + d)n , with c a constant that does not depend on j. Then, from (3 . 18) we have

card(NJ ) 'Loo cm(s + 1 + d)n C < . ( ) 2 ' - q3 8=1 1 + s m

Here, the summation is finite because m > 1tn , then C :::; car:�Nf) C(m), with C(m) a constant that depends on m . Finally, by Lemma 3 .6 , C -t O when j -t oo . It only remains to prove that for each i = 1 , ' " , T, R1; -t 1 <A (0) l 2 . Fix i , then

I r 'Pi (X - k)dX l 2 = I f 'Pi (x)dx - f . 'Pi (X)dX I 2 J Ai Q. Jan J(AJ Q.)c-k

:::; ( l $i (O) 1 + f l 'Pi (X) l dX) 2 J(AiQ.)C-k

= l$i (O) 1 2 + 2 1$i (O) 1 f l 'Pi (X) l dx+ " J(AjQ.)C-k

+ ( ( . l 'Pi (X) l dX) 2 J(AJQ.)C-k

Summing over f 1 ,; and dividing by q; , we have:

(3 .22)

1; 'L I l 'Pi (X - k)dX l 2 :::; card�l ,; ) l$i (O) 1 2

q kerl ,i J AJ Q. q

+ 2 1$i (O) I � L f l 'Pi (X) l dx q3

kerl ,j J(Ai Q.) C-k

+ � 'L ( f l 'Pi (x ) l dX) 2 q3

ker . J(Ai Q.)C_k 1 ,J Now, by definition of f1 ,i and Lemma 3.6 c) , it can be shown that Ki +k e (Ai Q)O e Ai Q for k E f1 ,i ' Hence (Ai Q)c - k e Kf. From this and property (2 .4) , we have:

1; L f . . 1 'Pi (X) l dx :::; � L f l 'Pi (X) l dx

q ker . J(AJ Q.)c-k q ker . J Kf 1 ,3 1 ,] < card(f1 ,j ) Cm

qi ( l + Mi)m ' Rev. Un. Mat. A rgentina, Vol. 44- 1

Now, consider e > O , using (3 . 14) we have that card�;,,j) < e + 1 for j large enough. Moreover, in the definition of Ki , we can choose a constant Mi such that ( 1+cM; )m < e

(for a fixed m) . Then, we will have: � L 1 l 'Pi (X ) l dx < c . Using this in qJ kEr' ,j (Aj Q)C_k

(3 . 22) we can conclude that for e > O and j large enough:

(3 .23) lj L 1 r . 'Pi (X - k)dX l 2 < Icí\ (OW + c . q kEr'j J AJ Q

On the other hand:

I r . 'Pi (X - k)dX l 2 = 1 r 'Pi (x)dx - 1 . 'Pi (x)dX I 2 JAJ Q JITtn (AJ Q) C-k

2': ( liPi (0) 1 - 11 . 'Pi (X)dX I ) 2 (AJ Q)c-k

2': l iPi (OW - 2 IiPi (0) 1 1 1 . 'Pi (x) dx l · (AJ Q) c -k

Remember that from (3 . 14) we have that 1 - e < car�;" j ) and

� L,kEr" j f(Aj Q) C-k l 'Pi (X) l dx < c . Then, summing over fl ,j and dividing by qj : lj L 1 r . 'Pi (X - k)dX l 2 2': card�l ,j ) liPi (OW q kEr' ,j JAJ Q q

- 2 IiPi (0) 1 � L 1 1 . 'Pi (X )dx l qJ ka',j (AJ Q) C-k

> ( 1 - C) liPi (OW - 2 IiPi (0) l c (3 .24) > l iPi (OW - c. Finally, from (3 . 23) y (3 . 24) , we conclude that for i = 1 , . . . , r, Rlj � liPi (0) 1 2 when j � oo .

D

3 .4 . PROOF OF THEOREM 3 . 1 . The proof of this result , is a direct consequence of Propositions 3 .2 , 3 .4 and 3 .5 . Suppose that 'P generates a MRA (Vj )jEZ . Then Properties (P 1 )- (P5 ) of the Íllultiresolution analysis are satisfied. Statement a) of the theorem is an immediate consequence of Property (PI ) and Proposition 3 .2 . Using Properties (P4) , (PI ) and Proposition 3 . 5 , then b) is verified . Now, suppose that 'P verifies a) and b) of the theorem. To prove that 'P generates a MRA, define Va = span{'Pi ( ' - k) }kEZn , i= l . . .T and Vj = {g(Ajx) j 9 E Va} · Then Property (PI ) is a consequence of a) and Proposition 3 . 2 . Properties (P2 ) is trivial due to the definition of Va and Vj , and (P5 ) is assumed . Property (P3 ) is a consequence of the hipothesis of orthonormal lattices translates and the localization property of each 'Pi as was proved in Proposition 3 .4 . Finally from b) and Proposition 3 .5 , Property (P4) is satisfied . Hence, 'P generates a multiresolution analysis with multiplicity r associated to the dilation matrix A and the lattice f.

Rev. Vil. Mat. A rgentina, Vol. 44- 1

88 A . BENAVENTE AND C. A. CABRELLI

o

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Numer. Math . , 73, ( 1996) , no. 1, 75-94. R. Horn y C. Johnson, Matrix Analysis, Cambridge University Press , 1 99 1 . J . Hutchinson, Fractals and Sell-simílarity, Indiana Univ. Math. J . , 3 , ( 1 98 1 ) , 713-747. R.-Q. Jia and Z. Shen, Multiresolutíon and wavelets, Proc. Edinburgh Math. Soco 37, ( 1 994) , 271-300.

[JRZ99] R.-Q. Jia, S. Riemenschneider and D.-X. Zhou, Smoothness 01 multiple refinable functíons and multiple wavelets, SIAM J. Matrix Anal . Appl. , 2 1 , (1999) , no. 1, 1-28 (electronic) .

[MaI89] S . Mallat Multiresolution ripproximations and wavelet orthonormal basis 01 L2 (R) , Trans . Amer. Math. Soc . , 3 1 5 , (1989) , 69-87.

[Mey92] Y. Meyer, Wavelets and Operators, Cambridge University Press, Cambridge, ( 1 992 ) . [Pot97] A . Potiopa, A problem 01 Lagarias and Wang, Master's thesis , Siedlce University, Siedlce,

Poland, June ( 1997) , (Polish) . [Shen98] Z. Shen, Refinable functíon vectors, SIAM J. of Math. AnaL , 29, ( 1 998) , 235-250 . ·[Woj97] P. Wojtaszczyk, A Mathematical Introductíon to Wavelets, Cambridge University Press,

( 1 997) .

INSTITUTO DE MATEMÁTICA APLICADA , UNIVERSIDAD NACIONAL DE SAN LUIS , EJÉRCITO DE Los ANDES 950, 5700 SAN LUIS , A RGENTINA , AND CONICET , ARGENTINA E-maíZ address, Ana Benavente: abenaven0unsl . edu . ar

DEPARTAMENTO DE MATEMÁTICA , FACULTAD DE CIENCIAS EXACTAS y NATURALES , UNIVERSIDAD DE BUENOS AIRES , CIUDAD UNIVER.SITARIA , PABELLÓN 1, 1 4 2 8 CAPITAL FEDERA L , ARGENTINA , A N D CONICET, ARGENTINA E-maiZ address, Carlos A . Cabrelli: cabrelli0dm . uba . ar Recibido : 8 de Agosto de 2002. Aceptado : 1 2 de Febrero de 2003 .



THE RESOLVENT ON IR" , A DEGENERATING METHOD.

CYNTHIA WILL

ABSTRACT . L e t G = S O ( n , 1 ) and K = SO(n) . We use a c o n t i n u o u s family o f

Lie algebras isomorphic to t h e L i e algebra of G , g , t.o degenerat.e the hyperbolic real space Hn c:::' G/K into the Euclidean space IH:n. . This al l ows li S 1.0 recover the resolvent of the Laplacian on IH:n from the resolvent of the hyperbol i c

L aplaci an .

l . INTRODUCTION

8 9

Let Hn = G/K be the hyperbolic space , where e = S O (n , 1 ) and K = SO(n) . By using the Cartan decomposition of the Lie algebra of e , 9 = e EB jJ , we change continuously the Lie bracket on 9 in order to make it vanishes on p , and so degenerating 9 to the Lie algebra 90 of the group of isometries of IRn . We then have a continuous family of Lie algebras 9s ( for s E [O , 1 ] ) , from 91 = 9 to 90 = e EB lE.n such that for s -¡:. o , 9 s is isomorphic to 9 . For s i- o , the t echniques we have in the hyperbolic case (see [5] ) are applied to calculatc thc radial part of the Casimir element of U (9s ) , and also we characterize the sphericaI functions as certain solutions of the following ordinary differential equation (note that i t also depends on s ) ( d2 d ) .

dt2 + s (n - 1 ) coth(st)

dt - As (V) f ( as ( t ) ) = O . ( 1 )

With al! this , we obtain an express ion for the resolvcnt kernel , given by a solution of the aboye equation with appropriate asymptot ic behavior . Then by looking at

the limits we obtain an express ion for the kernel in 90 . S t udding the differential equation this kernel satisfies , we also can express i t in terms of special functions (see ( 1 5 ) and ( 1 6) ) .

Although an express ion of the kernel of the resolvent of the Laplacian in this case is of course known , we think that this rnethod may be used in other cases where one knows something about the resolvent , as for example the other rank one symmetric spaces of noncompact type , or more general!y, the D arnek-R.icci spaces . AIso this approach might present sorne interest in considering the heat Kernel instead of the resolvent .

2 . P RELIMINARIES We begin by introducing notation that will be used throughout this papel' . As

is customary, we will denote a Lie group by an upper case letter and its Lie algebra by the corresponding lower case gothic letter .

Lct G = SO(n , 1 ) be the Lie group of matrices in SI (n + 1 , lE.) leaving the quadratic form - xI -x§ - . . . -x;, +X;,+ l invariant . Consider 9 = e + p the Cartan decomposition

Partially supported by a C O N J CET fellowshi p an e! research grants from SeCyT U N C , CONl C ET and F O N CyT ( A rgenti na) .

Rev. Un. Mat. A /gentina. Vol. 44- 1

90 CYNTHIA WILL

of its Lie algebra g, associated to the Cartan involution e (X ) = JX J , where J = [ -¿el � ] . Thus

e = { [ � � ] : A E SO (n) } and p = { [ _Obt � ] : b E �n } . AIso, if we put

Ho = L1 0 1 ] E P i t is easy to see that a = �o is a maximal abelian subalgebra of p . Let B be a multiple of the Killing form of 9 such that B(Ho , Ho ) = 1 , and we take the inner product ( - ; . ) given by Bo = -B( · , e · ) . Let {a} be the corresponding �ystem of positive roots of the pair (g, a) , such that a(Ho ) = 1 , and as usual , let p = n21 a.

We will identify the complexified dual space a� with e under the correspondence v = za f--t z. In other words , since a(Ho ) = 1, we are identifying v E a� with v (Ho ) .

For each s E [0 , 1 ] let eps : 9 f--t 9 b e defined by

eps (X + Y) = X + sY X E e, Y E p . Define g s = (g , [ ' , ' ] s ) the metric Lie algebra with underlying inner product space (g , ( - , - ) ) and Lie bracket given by

[X, Y]s = ep;l [epsX, epsY] , X, Y E g . Thus , ep8 : g8 f--t 9 is a Lie algebra isomorphism. It is easy to see that if go =

lims>-to g8 ' then go � t EB IRn is the Lie algebra of Mo (�n ) , the groQP of isometries of �n .

For each s E [0 , 1 ] ' let Gs denote the connected Lie group with Lie algebra g8 ' It is easy to see that

commute , that is

�. d gs -- 9 an

exp. ! � ! exp

Gs � G

ael . ( ) 9s - 9 [ 9 s exp. l ! e x p

Gs � Gl ( gs )

�s o exps = exps oepa and Ads o exps = e o ad.� . (2 )

On the other hand, by the definition of gs we have that for H E as ( = a V s ) , [H, X]s = ep; l [sH, epsX] . Therefore, if Xa E ga , and we take Xa. = ep; l Xa , then we have that [H, Xa. ] s = sa(H)Xa • . This implies that as = a o eps = sa is the restricted root (system) of the pair (gs , as ) . We have also that the corresponding Ps is given by Ps = s (n2- 1 ) due to our identification of a� .

If for each s , At = {exps (tHo ) : t > O } , we have the polar decomposition of Gs , Gs = KCl(At )K . We take on As , the Lie subgroup of Gs with Lie algebra as , the measure da · = dt ; on K we normalize the measure so that the total mass is one, and for the measure on Gs we have the following observation . Lemma 2 . 1 . The Haar measure on Gs relative to the polar decomposition is given by dx = Js (a )dk1 da dk2 • More precisely, if f E Cc (Gs ) ,

¡ f (g)dg = 1 . f (k1 ak2 ) Js (t )dk1 dadk2 G. KAt K


THE RESOLVENT ON ]Rn, A DEGENERATING METHOD.

:1 s i n h ( s t ) ( ) n - l whp1'e .]s ( o ) = s ' , fo1' a = exps ( tHo ) .

9 1

PTOof. The proof o f this fact is basically given in [ 1 , pp . 73] , and we wi l l on ly need the followillg two observatiolls when s appears in the proof.

• Since Gs = � exps (p s ) , we have that (see [1 ) pp .72 )

dx = det [ sin�

d�(;

)(Z) ]

dk dz where x = k exps (z ) .

• The function X � [X , H) induces an isomorphism from �/m into a; e p . l t i s easy t o see that its determinant i s given by ca (H)n- l

, where e is a nonzero constant independent of H.

Then we have that .]s (exps (tHo ) ) i s a multiple of

det [ s inh ads ( Ad (k) tHo ) ] a(tHo )n- l = det [Ad(k)

sinh ads (tHo ) Ad ( k ) - l ] tn- 1 ads ( Ad(k ) tHo ) ads (tHo )

as asserted . o For simplicity we will denote Js (t) = Js (exps (tHo ) ) .

Definition 2.2. A function f on G is called K- biinvariant o r radial i f f (kxk' ) = f (x ) for all k , k' E K.

Let C ( G s // K) be the space of continuous radial functions , and we will also denote by C= (Gs //K) and Cr;;o (Gs /IK) , the space of smooth radial functions and the compactly supported smooth radial functions , respectively.

Let f - denote the restriction to At of a function f E C( G s // K) . lt follows Trom the polar decomposition Gs = KCI (At )K that f is determined by f - . Moreover , if D is a differential operator on G s invariant under the right and left action by elements of K , we can define the radial component of D , as a differential operator on At such that (see [ 1 , § 4 . 1) )

(D 1 )- = 6. (D) f- , v f E C= (Gs // K ) ) . (3) Let C denote the Casimir element of the complexification of 9s , (9s ) e , wi th

respect to B. We are interested in the radial component of C . To calculate i t , we will use arguments analogous to those in [8 , pp. 280) .

Let X1 , . . . , Xn- 1 , be a basis of 9(}' , such that - B (Xi , B (Xj ) ) = 6i,j , i . e . orthonormal with respect to ( , ) . For i = 1 , . . . , n - 1 , consider Xt = cjJ-; 1 Xi . Thus , from the previous observations we have that Xt E 9 (}' , for al! 'í , lf we define , as usual , for j = O, . , . , (n - 1 )

Zj = 2 - � (Xj + B(Xj ) ) , 1j = 2- � (Xj - B (Xj ) ) , and the corresponding ZJ and Y/ , it is easy t o see that ZJ = Zj and Y/ = s - l yj . On the other hand , since the inner product do es not depend on s , we have that in each 9s tbe Casimir operator is given by

E U (9s ) .

Tberefore thc action of Cs on C= (G s // K) is given by the following ordinary differential equation .


92 CYNTHIA WILL Proposition 2 . 3 . JI I E C'X (G,, // K) then

( d2 d ) Cs / (a .. ( t ) ) = 2" + s (n - 1) coth(st) -d f (as (t) ) .

dt t

In particular , we have that in the limit

( d2 n - I d ) lim Cs f (as ( t ) ) = 2" + ---d f (tHo ) · S""" o dt t t ( 4 )

which i s the action of the Laplacian on IRn acting on radial functions ( see [2 , pp. 266) ) .

Proa/. From the definition of the Z¡ and ( 2 ) we have that

Ads (as ( - t ) ) Zi = Ads (as ( - t) ) Z¡

Thus

cosh(st) Z¡ - sinh (st) Yis

= cosh (st) Zi - sinh (st) s- l Yi .

y; = s coth(st) Zi + s sinh -1 (st) Ads (as (-t ) ) Zi ,

and then we get

�2 = s2 coth2 ( s t ) zf + ..,2 sinh -2 (st) ( Ads (as ( -t) ) Zi f _ S2 coth ( 8t) sinh - 1 ( st) ( Zi . Ads (as ( -t ) ) Zi + Ads (as (-t ) ) Zi . Z, ) .

Note that (5 ) also implies that

[Zi , Ads (as ( -t ) ) Zi] = _ 8 - 1 sinh (st) [Zi , YiJ , and then we have that

y? = 82 coth2 (st ) zf - 82 sinh-2 (st) [Ads (as ( -t ) ) Zi] 2

-282 coth(st) sinh - 1 (st) Ads (as (-t) ) Zi . Zi + S coth(st )Ho .

Therefore if f is biinvariant for the action of K , we have that

and then

as was to be shown .

d Zi f (g) = -

d f (g exps (tZi ) ) = O ,

t I t=o

Yi2 f (as (t ) = s coth(st)Ho f(as (t) ) ,

Rev. Un. Mat. A rgentilla, Vol. 44- 1

( 5 )

o

THE RESOLVENT ON ]R1l, A DEGENERATING METHOD.

3 . S P H ERI C A L F U N CT I O N S AND THE RESOLVENT 93

Definition 3 . 1 . If cp is a complex valued radial continuous function on cs , then cP is said to be a K - spherical (or s imp ly sJlhel'ical) function if cp ( e) = 1 and Gs cp = Acp for some A E e.

For each 8 E [0 , 1 ] ancl lJ E e, let CPs ( IJ, ' ) be the spherical function on Cs with eigenvalue As (11) = 112 - p; . Remark 3 . 2 . We note that this eigenvalue is notarbitrary, and in order to calculate it for 8 =J 0, sin ce 9s is isomorphic to g , one can proceed exactly as in SO(n, 1 ) to see that such functions are given by a matrix entry of the spherical principal series of Cs associated to the character x� (man) = aV+P• (see [ 1 , pp . 103] ) .

We then have , for each s =J O , that CPs (II, · ) is the solution of the following differential equation ( d2 d )

dt2 + s (n - 1 ) coth (s t ) dt - As (11) f(as (t) ) = °

continuous at t = O , and such that f ( a s ( O ) ) = 1 .

(6)

Note that since lim t8 (n - 1 ) coth(st ) = n - 1 , this equation has a regular singular tHa point at t = O . Moreover , i f we set z = st i t is easy to see that f (z) satisfies the equation (6) if and only if

or equivalently

( d2 d ) 82

dz2 + s2 (n - 1 ) coth (z ) dz - As (11) f (z) = 0 ,

( 7)

This is a Jacobi equation with parameters A = i � , a = n22

, f3 = - � , and I = P (see [3] ) , and therefore , we have that for each s =J 0 , the spherical functions r!Js are given by Jacobi functions in the following way :

( n - 2 _ 1 ) CPs (lI, a8 ( t ) ) = 'P ,; 2 • 2 (st) .

Equivalently, in terms of the Gauss hypergeometric functions we have that

( ( ) ) ( PS - 1I Ps + 1I . ' 2 ( )) cPs 11, as t =2 Fl � ; � ; nj - smh st .

It can al so be seen (see [3 , pp . 7] ) that for s =J ° and 11 ct -sN, a second solution of the equation (6) in (O , +(0) is given, in terms of the hypergeometric function, by

- - ( "- +p) ( ps + 1I 8 (n + 1 ) + 211 11 2 ) Qs (lI, t ) = (2 COsh(8t) ) ' 2 Fl � ; 4s j � + 1 ; cosh- ( s t ) .

If Re 11 > ° , the asymptotic behavior of these functions as t -7 00, is given by :

where

cp(lI, as ( t ) ) = c ( lJ, s ) el ( IJ ..,-p, ) , Qs (lI, t ) '" e- ( v+P. ) t ,

2P- '; r ( � ) r ( � ) c( II S ) = 8 _ ' " r ( p� + /J ) r( s (n+ l ) +2v ) ' 2s 4 8

Rev. Un. Mat. A 1xentil/o. Vol. 44- 1

94 CYNTHIA WILL

, It is proved in [5) (see also [6] ) that fol' s = 1 , the function Q 1 (1/, al (t ) ) = ;:�¡�:?) is a solution of the equation (6 ) (with s = 1 ) , such that

lim ,Ir ( t) Q I ( v , ([ I ( t ) ) = 0 , t>--tO+

, d hm JI ( t) -d (J I ( v, a l ( t ) ) = 1 , t>--tO+ t

and moreover , it is also proved that the resolvent of the Laplacian on GI l K (in certain half plane of <C) is given by convolution with the K-biinvariant function on G extending this function. Analogo\lsly we can generalize this , in the following theorem, for the other values of s . Theorem 3.3 . For each s E (0 , 1 ) and v E e , 1/ � -sN, there exists a function Q s (1/, . ) E Cco (G s - K // K) with the following properties:

(a) CsQs (v, ' ) = .-\(v, 8) Qs (l/, . ) , d

(b) lim Js (t) Q s (v, as (t) ) = ° and l im Js (t) -d Qs (l/, as (t) ) = 1 . t>--tO+ t>-4O+ t (c) Where defined, Qs (l/, g) E Ltoc (Gs ) , and if Re l/ > Ps , Qs (l/, g) E V (Gs ) ' (d) Jf f E Cgo (G8// K) and 1/ � - sN then

r Q8 (1/, X- 1 y ) (C8 - '\ (1/, s ) Id ) f(y) dy = f (x) . (8) Jos Proa/. The proof of this theorem is' essentially the same as that of Theorem 2.2 in [6) (see also the references given there ) and therefore we wil l just make sorne observations .

Let Q8 (V, · ) be a K-radial function on Gs such that restricted to As is given by

08 (1/, t ) S2p Qs (l/, as ( t) ) = - ( ) ' 21/C 1/, S It is easy to see from the aboye remarks that this function satisfies (a) . To see (b) , recall that this function is a solution of equation (6) and this equation has a regular singular point at t = O . The corresponding indicial equation is given by a (a + (n - 2) ) = 0, with solutions a = ° and a = - (n - 2) . It is clear that 4>s (I/, · ) is the solution corresponding to a = O. If 1/ � -8N, we know that 08 (1/, t) is a linearly independent solution , and therefore , by the general theory of regular singular points , we have that , when t f-t ° Os (l/, t) "-' ds (l/)tn-2 1 Iog(t) 1 6 n . 2 where ds (l/) is a meromorphic function of v. Hence, lim J8 (t)Q8 (V, as (t ) ) = O .

t>--tO+ J' ( t ) Finally, to prove the second part of (b) , we note that s (n - 1 ) coth(st) = J.tt

(see Lemnia (2 . 1 ) ) . This fact gives us an analogue of formula (*) in [5) pp. 669, and then we can proceed as in [5 , lemma 1 .3) . O

4 . T H E RESOLVENT KERNEL

We first note that part (d) of the aboye theorem implies that in the limit , for Re (l/) > 0 , (� - v2 Id) - 1 is given by convolution with QO (I/, ' ) , where Qo (I/, · ) is a radial function on IRn such that Qo (1/ , t) is the solution of the following differential equation (see (4) ) ( d2 n - 1 d '» )

dt2 + -t - dt - 1/- f (t) = 0 , (9)

such that

THE RESOLVENT ON Rf1 , A DEGENERATING METHOD.

limJo (t)Qo (v, t ) = O and lim Jo (t ) dd Qo (v, t ) = 1 .

tHO 1,>-l O t Here Jo (t ) = (2t )n- 1 .

95

In order to have explicit solutions of (9 ) , we will introduce now the Bessel equations . The following differential equation

1 (

7}2

) u" (t) + -¡u' (t) + 1 - t2 u(t ) = O

is called a Bessel equation . It is easy to see that this equation is equivalent to

r (t) + 2a: 1 t(t) + (¡.t2 _ r¡2 � a2 ) f (t) = O ,

where u (t) = tO: f (¡.t- 1 t) .

( 10)

( 1 1 )

Therefore , i f we take r¡ = a = n;-2 and ¡.t = iv then we have that the equation (9) is equivalent to the Bessel equation

u" (t) + �u' (t) + (1 - ( n ;, 2 ) ') u (t) � 0 ,

where u (t) = t n ;2 f (-Iv ) .

( 12)

The solutions of this differential equation are well known . We will now summarize some known results on them, following [7, Ch.3 , sec 6] .

First , let Jr¡ be the Bessel function . It is given by

Jr¡ (Z) = [r (�) r (r¡ + �)r l ( �r i

l

1( 1 - e )r¡- � ez ti dt .

Note that the integral in the above formula is meromorphic in r¡ with simple poles at r¡ + � E - N, and these poles are cancelled by the factor r(r¡ + � ) - 1 . Thus , these functions give smooth solutions of the Bessel equation , analytic in r¡. We also have that for r¡ = k + � this integral can be calculated explicitly, since it involves ( 1 - t2 )

k , and then Jk+ � is an elementary function for each k E N.

Other type of solutions of ( 10) are given by the so called Hankel functions , 1{�1 ) and 1{�2) . These functions form a basis of the space of solutions of ( 10) and they are linearly independent with Jr¡ . One can see that for Re r¡ > - � and 1m z > O , 1{�l ) is given by

-

11.{ 1 ) (z ) = . - (t2 - 1 ) r¡- � eiz t dt . 2e-71"ir¡ (Z)r¡ loo

r¡ l7l'r(r¡ + � ) 2 1

We also have that for r¡ = k + � , 1{k+ 1 are elementary functions for k E N. . 2 Finally, if we let

JCr¡ (r) = � (� ) '1 100 t- 1 - r¡e f.- - t dt ,

it can be seen that it is a solution of ( 10) , and in fact (see [7] pp . 233) , for r > O we have that

JC'I (r) = �7l'ie71"¡ '1/ 21{�1 ) ( i1' ) . ( 13 )

Rev. Un. Mat. A /:f?entinCl, Vol. 44- 1

96 CYNTHIA WILL

We then have that any solution of (9) is given by

fv (t) = a C �+l J� - l (ivt) + b C �+l H�� l ( ivt ) ( 14)

for some a , b , E C. From the properties of Jr¡ and H;/ ) listed aboye , i t c a n be seen that 1Jo (v , t ) is a constant multiple of

(vt) l - � J 1!. - 1 (ivt) . 2 On the other hand, in order to have an explicit expression for the resolvent

kernel , we first note that since the term in ( 14) corresponding to Jr¡ leads to a Coo eigenfunction of .6., \\re could set

Qo (v, t) = b C �+l H�� l (ivt) . 2

We also have that the asymptotic behavior of H�� l as ( H O is given by 2

'1/ ( 1 ) ( . t ) '" _ . "2 -� 2 r( n 1 ) ( ) 1!. - 1

TL 1!. - 1 I V 1 • , 2 � lvt and therefore , by straightforward calculation we have that

t �+ l ( 2 ) l - � �i ( 1 ) • Qo (v, t) = - r(� ) iv 2H � - 1 (lVt) .

It is easy to see fram ( 13) that the aboye formula is equivalent to

21 - � ( t ) l - � Qo (v, t ) = - r( � ) -;; K� _ l (vt) , t > O .

Recall that if n is odd, 1-l��1 is an elementar y function. 2

( 1 5 )

Remark 4 . 1 . We would like to point out that the difference between ( 1 5 ) and the corresponding formula [7, (6.49) pp. 232] (see also [4 , ( 1 .26 ) pp . 7] ) , is due to the fact that in our case the Haar measure on K is normalized ( i . e . K has total mass 1 ) . It is easy to see that with this normalization , we would have to consider the kernel Ro (v, t) = YOldn- l ) Qo (v, t) , where Vol (sn- 1 ) = �(:) ' and then we obtain

Ro (v, t) = _ (2�) -n/2 (�) l - � K� _ l (vt) ( 16)

as in [7 , (6 .49) pp. 232] .

REFERENCES

[ 1 J G A NGOLLI R . , YARADARAJAN V . , Harmonic analysis of spherical functions on real reductive groups, Spri nger VerJag, New York, 1 988.

[2J HELGASON S . , Groups and Geometric A nalysis, Pure and Applied M at h 113, Academic

Press , 1 984 .

[3J KOORNWINDER T . , Jacobi functions and analysis on non compac t s emisimple Lie groups, 'Special functions: Group Theore tical A spects and Applications ' pp 1 -8 5 , Reidel , Dordrech t ,

1 98 4 . [4 J M E LROSE M . , Geometric Scattering Theory, Stanford Lect . , Cambridge U n i versi ty Press ,

1 995 . [S J M I ATELLO R . J . , WA LLACH N . R . , The resolvent of t h e Laplacian on negaüvely c u rvcd local/y

symrnetric spaces of finite volum.e, Jour. Diff. Geometry 36 ( 1 992) , 66:1-698 . Rev. Un. Mat. A rgentina, Vol. 44-1

THE RESOLVENT ON ]R11 , A DEGENERATING METHOD.

[6] M I ATI·: J . LO R . , WILL e . E . , The residues of the resolvent on Damek- Rieei spac�s. P roceecl i tl g� o i' ÜI" A 'vI . S , vo l . 1 28 , 2000 , 1 221 - 1229.

[7] TA Y L O H M . , Partial Differential equations l . Springer Verlag . , New Yor k , 1 996 . [8] WA L L A C H , N . , Harmonie A nalysis on Homogeneous Spaees , (Pure and A pp l . ivl at h . 1 9 ) .

Maree[ Dekker ine . , New York , 1 973 . DEPA H' J ' M ENT O F M AT H E M ATICS , YA LE U N IVERSITY , 1 0 HILLHOUSE B ox 20828 :1 "' EW H AV E N .

CT 0 6 .520 l ' S A ( C U H H E N T A F F I L I AT I O N ) E-maíl adrl'ress : cynthia . w illlDyale . edu

FA M A F A N D C I E M , U N IV ERSIDAD NACIONAL DE CÓRDOBA , 5000 C Ó R D O B A , A H G E N T I N A E-mail addn�ss : cwilllDmate . uncor . edu

Recibido : 23 de Agosto de 2002.

Aceptado : 1 0 de Marzo de 2003 .

97


REVISTA DE LA UNION MATEMATICA ARGENTINA Volumen 44, Número 1 , 2003, Páginas 99- 1 08

The Adaptive Dynamics For The Randomly Alternatmg Prisoner' s Dilemma Game

Esam EI-Sedy*

99

Department of Mathematics, FacuIty of Science (48), King Khaled University, Abha, PO Box 9004, Saudi Arabia

Abstract.

We consider a game with two players and two choice s for each one. In each round of

the aIternating model, there is one option for one player called leader. The two players

have the same chance to be leader. In this model each two consecutive rounds represent

one unit . We consider strategies realized by simple transition rules depending on the

previous outcome. We consider homogeneous population of strategy S, and ask for the

most favorable adaptation. Any parameter x in S changes according to the adaptive

dynamics � = iF ; F is the payoff for S-player. a l .Introduction.

Many examples of reciprocal aItruism are modeled by an aIternating Prisoner' s

Dilemma (PD) game see [ 1 1 ] . I n this game w e have two players 1 and I I and two choices

e to cooperate and D to defect . In each round of the game one of the two players

choose his option to defect . In each round of the game one of the two players choose his

option and the other player reply with his option in another round . And this means that

for each round there is a single option for one of the two players. This player is called

leader (or donor) and the other is called recipient . Consequently the leader control what

the outcome is going to be .

The two players in this game aIternate the leader role and both of them have the

same chance to be leader. There are two types for this game, strictly aIternating where

the two players exchange the leader' s role every round and the randomly aIternating

where the two players exchange the leader' s role randomly. In this paper we shall be

interested in the randornly aIternating PD game. 1

1 99 1 AMS Mathematics Subject Classification : 90A. 90D, 62C, 62P Key words and phrases : Adaptive Dynamics. Alternating Prisoner's Dilemma Game, Transition matrix *e-mail address : esam_elsedy@hotmail .com


1 00 ESAM EL-S EDY

2-The Transition Matrix For The Game.

According to the evolutionary game dynamics, each player receives points to

represent the increments in fitness see [8] . For the randomly aIternating PD game, leader

decides between the two choices e and D option e yields a points to the donor and 13 points to the recipient, while option D yields X points to the donor and O points to the

recipient . In a single round, option D is better than e for the leader. We shall as sume

that the cost to the donor is less than the benefit that brings to the recipient see [3 ] . The

loss is X - a and the benefit is 13 - O , then we have

O < x - a < 13 - 0 ( 1 )

Now, we consider two consecutive rounds in which the players exchange the leader role

in turn. Ifboth play C. both earn a + 13, which we denote by R, if both play D, both

earn X + O, which we denote by P, but if one plays e and the other D, then the

cooperator earns a + O which we denote by S and the defectors earns X + 13 which

we denote by T. From ( l ) follows

T > R > P > S (2)

and

2R > T + S (3 )

The inequalities (2) and (3) are usual conditions for the payoff for the simuItaneous PD

game. Adding, we obtain

T + S = P + R (4)

Conditions (2), (3) and (4) describe the alternating PD game, while condition (4) means

in the simultaneous PD that the cost from switching D to C is the same against a

defector as against a cooperator (P- S = T- R) . The four states of outcomes a. 13, X and

O represents the possible payoff obtained by one of the two players for one round. If we

denote these out comes byl , 2, 3 and 4 resp . , then the possible strategies for each player

can be represented by the quadruple (11 1 . 112. 113. 114), where II¡ denotes the probability to

play C after outcome i. These probabilities are independent of the random decision of

who is going to be leader. If a player with strategy P =(PI . P2 . P3 . P4) and probability

y to play X in the first round match against a pI ayer with strategy º = (ql . q2. q3. q4) and probability j to play e in the first round, then the transition matrix for the match

from one round to the next is given by

THE ADAPTIVE DYNAMICS 101

(5)

For instance the transition probability from state 1 to state 2 for the P-player equals the

product of the probability tha! the Q-player is leader, which equals ..!.. and the probability 2

q2 that the Q-player cooperates after receiving f3 in the previous round .

3-The PayotT of a Player in Randomly Alternating PD Game.

If (O denote the probability that the game proceed to the next round, then we have

. two cases to study, when (O = 1 (limiting case) and (O < 1

1 -For (i) = 1

The stationary distribution of transition matrix M is the eigen vector n where

4 n = (JiI , Ji2 , Ji3 , Ji4 ) ; ¿ Ji; = 1 ,

;= 1

which correspond to the left eigenvalue of M see[ 1 2] . Hence n is given by

n M = n (6)

4 under the condition ¿ Ji ¡ = 1 , we get

Í= I

Let Ji 1 = Ji and Ji 2 = Ji' , then from (7), we get

n ( , 1 1 ' ) = Ji Ji - - Ji - - Ji

" 2 ' 2

Solving (6) and (7) in Ji, Ji' yeild

(p) + P4 )(2 + q} - ql ) - (q3 + q4 )(P4 - P2 ) Ji = --��--��--�------��--2 [(2 + P3 - PI )(2 + q3 - ql ) - (P4 - P2 )(q4 - qJ]

;r' = (q3 + q4 )(2 + p) - pJ - (P} + P4 )(Q4 - Q2 )

2 [(2 + P3 - PI )(2 + q3 - QI ) - (P4 - P2 )(Q4 - Q2 )]

Now, if F is the payoff for the P-player, then F is given by

(7)

(8)

102

Using (8) we get

ESAM EL-SEDY

1 (a- x) + 7r' (P - o ) + - (:x; + o ) 2

F=� (xi-8)+ (a-x)[( 2+v )- t��)[t( 2,+U )-r ,tl]

2 2 [( 2+4( 2+v)-.u .u ] (9)

where U = P3 - Pl ' .u = P4 - P2 ' r = P3 + P4 , and similarly for u' , /-i , and r' . In this

case we note that the initial probability yand JI play no role .

2- F or CtJ . < 1 The total payoff F for the P-player against Q-player in this case is given by

1 F= - [yA +(1 -y)X+ y' B + (1 -y' )L\ ] 2 .

1 . = 2[X+L\ +,;{A-)Q+Y(B-L\)] ( l O)

where A, B, X and LI are the expected payoffs for the P-player, given that the first

round of the game resulted in a, P, :x; and o resp . .

Then w e have .'

( l 1 )

where .!. is the probability that P-player will be leader in this round. Equation ( 1 1 ) and 2

the corresponding equations for the other expected payoffs B, X and LI can be written

as

( 1 2)

where

A = / - aM ( 1 3 )

and / is the 4-unit matrix has full rank, so that we can compute the payoff. From ( 1 2) we have

THE ADAPTIVE DYNAM ICS

2 X + LI + PI(A-X) + q 2(B-Ll) + - (a-A) = O

(V

2 X+ LI + PiA-X) + ql(B-Ll) + - (J3-B) = O (V

2 X + LI + P;(A-X) + q4 (B-Ll)+ - (X-X) = O

(V

2 X + LI + P4 (A-X) + q3(B-Ll)+ - (o-Ll) = O

(V

1 03

( 1 4)

( 1 5 )

( 1 6)

( 1 7)

Solving equations ( 1 4L( 1 7) in (A -X). (B -Ll) and (X + LI). we get that (from ( 1 4) and

and ( 1 6»

(2 + (vu )(A-X) + (vu' (B-Ll) = 2(a-x)

and from ( 1 5 ) and ( 1 7) that

m,LL(A-X) + (2 + (vu' ) (B-Ll) =2(J3-0) Then using Cramer' s rule for ( 1 8) and ( 1 9) ,we get

A-X = 2 (a- V(2 + (vu ' ) - (J3 - O )cV,LL ' (2 + (VU )(2 + mu ' ) -(V 2,LL¡.i

and

B-LI = 2 ( P - o )(2 + wu) - (a.- x.)W,LL (2 + wv)(2 + wu ) - W2,LL¡J

Adding ( 1 6) and ( 1 7), we get that

1 (Vi mi ' X + LI = - [(X +0) + - (A-X)+ - (B-Ll)] 1 - m 2 2

( 1 8)

( 1 9)

(20)

(2 1 )

(22)

For the two cases of (V and, the previous computation, we get the following result . The

total payoff F for an S-player is given by ( 1 0) for O) < 1, where X + LI. A -X andB -LI are

as in (22) , (20) and (2 1 ) . And for O) = 1 , F is given by (9) .

4-The Adaptive Dynamics For The Alternating PD Game.

We consider a homogeneous population of S-players, and ask for the most favorable

adaptation. If an individual was permitted a small deviation from strategy S. which

direction would be most favorable (In a biological context, the small deviation would be

produced by a mutation and natural selection) . Now ifx is any parameter belonging to S,

where x can be y, PI . P2 . . . . . or P4, then x changes according to the adaptive dynamics, by

Rev. UIl. Mat. A ';r;entillll. Vol. 44- 7

1 04 ESAM EL-SED Y

oF x = -ax (23 )

where the right side of (23) evaluated at S=S '. The differentiation of F yields very

cumbersome computation, so we shall use the implicit function tlieorem. Equations

(1 2) can be written as

for instance

and see that

/; (PP P2 , P3 , P4 , Qp Q2 , Q3 , q4 , A ,B ,X ,d) = 0, i = 1 ,2,3,4

A = 0(11 , 12 , 13 , 14 ) o( A , B , X, d )

Hence, by the implicit function theorem, we have

o (A ,H ,X ,d ) {o {f¡ ,f2 ,h ,f4 )].1 o (f¡ ,f2 ,h ,h ) =�(A-X)E (24) o CA ,A ,A ,P4 ) o (A ,B ,X ,d ) o CA ,A ,A ,� ) 2

where

E = A -1

and from ( 1 3 ), we have

E = (1 - aJ M)-I = 1 + aJ M + (aJ M)2 + . . .

(25)

(26)

AlI elements eij of E are strictly positive. The formulae of these expressions (26) are

rather cumbersome and for futur use.

We note that

1 - aJ ' det A = - (2 +aJ(u+,u» (2 +co(u-,u»

.u (27)

and as one can easily deduce from the special form of A , So by straight forward

computation we can write E in the form

E = [l � ; �l (28)

Using (25) , we get


and hence

THE ADAPTlVE DYNAMICS

(O 2 a+b = -- (2+al(v-.u))(- -2+P3 +P4 )' 4 detA al

(O c + d = -- (2 + (0 (v - .u))(2 - PI - Po ), 4 det A -

(O e + f = -- (2 + (0 (v - .u))(P3 + P4 )' 4 det A

(O 2 g + h = -- (2 + (0(v - j../))(- - PI - po ) 4 detA (0 -

(29)

(30)

(3 1 )

(32)

al 2 (a + b) - (c + d) = -- (2 + c0(v - j../))(- - 4 + PI + P2 + 03 + P4 )' (3 3 )

4detA (O

al 2 (e +j) -(g+h) =--(2+ (O ( v - j../ ))( -- +A +A. +P3 +P4)' (34)

4det A al

Now trom ( 1 0) and (23 ), we get the following

• 1 Y = "l (A - X) ,

• 1 8A 8B 8X 8f! PI = - [y(- + -) + (1 - y)(- + -)] 2 Opl 8PI 8PI Opl

= (O [y(a + b) + (1 - y)(c + d)](A - X)

4

• 1 8A 8B ce 8f! P3 = - [y(- + -) + (1 - y)(- + -)]

2 Op3 Op3 8p3 Op3

= (O [y(e + f) + (1 - y)(g + h)](A - C) 4

. .

( 35 )

(36)

(37)

1 05

We note that the sign of PI and P3 is the same as the sign of A - C which has been

computed in (20) . Using the previous expressions together with the equations

(29)_(32), we get the following results .

Theorem. The adaptive dynamics for the randomly alternating PO game is given by

( 1 ) For co < 1 • 1 Y = - (A - X)

2 . . d 2 R = A = -

4 (l '

)( 2 + (O ( v + j../ )JI[ 2 - /-l-A+ Y ( - - 4 -tf� +A +A -lA )] (A-X) (3 8)

:-(0 (O

1 06 ESAM EL-SEDY

• • al 2 A=J.?¡ = -- e 2 + (O ( V + ¡t )t[ 2 - ¡.¡ -A+ Y ( - +¡.¡ +A +/1 +J.?¡ )] (A-X) (3 9) 4 0� ) w

where

A - X = 2 (a- x)(2 + wv ) - (P - o )w¡t (2 + (O (v - ¡t»(2 + (o (v + ¡t»

(2) For w = 1 (limiting case)

• • r[(a- x)(2 + v) - «(3 - o )¡t] PI = P2 = 2(2 + v + ¡t) 2 (2 + v - ¡t) ,

• • (2+v+¡t- r)[( a-x)(2+v)-(P-o)¡t] A =P4 = 2i..2+V+¡t)2(2+v-¡t)

For every case we note that

. . . . .

(40)

(4 1 )

(42)

( 1 ) PI = P 2 and P3 = P 4 and this means that the optimal adaptation depends only on

whether there was a e or a D in the previous round, and not on who actually

implemented it .

. . (2) Y and all Pi has positive sign and this gives the zone of cooperation, which is

defined by

(a - x)(2 + wv ) > ( P - o )(o¡t (43 ) and it i s independent o f y.

If a = P - o , which is just

T - P , and hence greater than one i. e a > 1 , we get from x - a T - R

(43 )(after substituting with u and J..l ) that the zone of cooperativity is given by

2 - < PI - P3 + a(p2 --: P4 ) w

Condition (44) implies to the following.

(44)

The zone of cooperativity is non empty if and only if it contains the Tit For Tat

strategy, which is given by (PI, P2, P3, P4) = ( 1 , 1 ,0,0), i . e if and only if the condition

2 - (0 a > -(O

holds. This condition agree with a > 1 .

In the limiting case O) = 1 , we see that condition (45) is always satisfied . .


(45 )

THE ADAPTIVE DYNAMICS 1 07

Cooperation is easier to achieve the larger of the zone, i .e the temptation T-R to defect

unilaterally is smaller compared with the gain R-P obtained by mutual cooperation.

In particular, if

2 a > co (46)

Then from (44), the zone of cooperation contains the strategy given by (P¡,P2,P3,P4) = ( l , l , 1 ,O), which is always ready to cooperate except if it has been played for a suker,i .e

if it has experienced a <> in the last round. In this case, it defects if it is leader in the next

round, but it defects only once.We note that in the limiting case co =1, condition (46)

simply means that the cost to the donor x-a is twice as large as the benefit to the

recipient �-<>.

It is easy to find the consensus strategy with highest payoff which irnmune to defection.

If all members of the population adopt this strategy, then the exploiters with a lower

propensity to cooperate cannot invade, and the overall payoff for the population is

maximal (subject to this non-invadability condition).This payoff is given by

REFRENCES.

F( ) - (a - l)(x - a)

(2 cor ) X + <>

s, s - y + -- + . 2(2 + co(v + ,u» (1 - co) 2(1 - co)

1 - Axelrod, R. , The Evolution Of Cooperation, Basic Books, New york, 1 98 8 .

2 - E l Sedy, E . and E l Shobaky, E. ,The perturbed payoff matrix of the repeated

Prisoner' s Dilemma game , Far East J. Appl. Math. 1 (3) ( 1 997), 255-270.

3-EI Sedy, E. and Kanaya, Z., The perturbed payoff matrix of strictly alternating

Prisoner' s Dilemma game, Far East J. Appl. Math. 2(2) ( 1 998), 1 25- 1 46 .

4- El Sedy, E . and Kanaya, Z., On Simultaneous repeated 2 D�ames, Revista de la

Unión Matemática Argentina vol.4 1 , 3, 1 999.

5- Hofbauer, J. and Sigmund, K., Dynamical systems and the Theory Of Evolution,

Cambridge University Press 1 988 .

6- Hofbauer, J. and Sigmund, K. , Adaptive dynamics and evolutionary stability o f Appl.

Math. lett. vol. 3, No 4, pp.75-79, 1 990.

7-Lindgrn, K., Evolutionary phenomena in simple dynamics, in Artificial lif II(ed.c.G.

Longtonet. al), Santa Fe Institute for Studies in the Sciences of Complexity X ( 1 99 1 )

108 ESAM EL-SEDY

295-3 1 2 .

8-Maynard Smith,J . , Evolution and the Theory o f Games, Cambridge University Press

1 982.

9-Nowak, M. and Sigmund, K. ,The evolution of stochastic strategies in the Prisoner's

Dilemma, Acta App. Math.20:247-265 , 1 990.

1 0-Nowak, M. and S igmund, K., Choas and the evolution of cooperation, Proc. Nat.

Acad. Science USA 90 ( 1 993a), 509 1 -5094.

l l -Nowak, M. and Sigmund, K., The altemating Prisoner' s Dilernma, J. Theor.Biol.

1 68 ( 1 994), 2 1 9-226 .

1 2-Nowak, M. and Sigmund, K. and El Sedy, E . , Automata repeated games and noise, J.

Math. Biol . 3 3 ( 1 995), 703-72

Recibido : 1 7 de Abril de 2002. Aceptado : 23 de Agosto de 2003 .


CRONICAS DE LA UI REUNION ANUAL DE LA D.M.A. 1 09

REUNION ANUAL DE LA UNION MATEMATICA ARGENTINA.

En homenaje a Luis Antonio Saútaló Las LII Reunión Anual de Comunicaciones Científicas, XXV Reunión de Educación Matemática y XIV Encuentro de Estudiantes, tuvieron lugar entre el 1 6 y 20 de septiembre de 2002, en la ciudad de Santa Fe de la Veracruz, organizadas por la Facultad de Humanidades y Ciencias, Facultad de Ingeniería Química, Imal y Ceride, de la Universidad del Litoral . La Reunión resultó exitosa, por lo que la Comisión Directiva quiere manifestar aquí su agradecimiento a la Comisión Organizadora Local y a sus colaboradores que hicieron posible que la Reunión Anual tuviera tan buenos resultados .

1 . 1 . - Se dictaron cursos de perfeccionamiento para docentes de nivel primario secundario y universitario, y cursos para estudiantes de licenciatura y profesorado . Un listado de los cursos es el siguiente:

1 1 1 - CURSOS PARA ESTUDIANTES DE LICENCIATURA . . CL ! Wavelets, Marcos y Teoría de SamplinR Carlos Cabrelli - (UBA) CL2 Geometría Simpléctica y Mecánica Hemán Cendra (UN Sur) CL3 Splines Ornar Roberto Faure (UTN,

Concepción del Uruguay) CL4 Grafos Marisa Gutiérrez (UNLP) CL5 Modelo Estructural de Series de Tiempo Orlando José Ávila BIas (UNSA)

1 . 1 .2 . - CURSOS SOBRE ENSEÑANZA DE MATEMATICAS

CE l Modelos matemáticos: Cadenas de Markov Néstor Aguilera (UNL)

CE2 Introducción a Geometrías No-Euclideanas Graciela Birman (UNCPBA) CE3 A ritmética y Criptografia Juan C . Canavelli (UNL) CE4 Exploración y presentación de datos . Un Elena F . De Carrera (UNL)

tema actual

CE5 Trigonometría para la EGB y el Polimodal. Eleonora Cerati e Ingrid Schwer Enfoques en la resolución de problemas y en (UNL) el modelado

CE6 Optimización Ma. T.Guardarucci (UNLP) CE7 Matemática Discreta para la EGB 1 Y 2 Bibiana Iaffei (UNL) CE8 Sistemas dinámicos y caos Gloria Moretto y Lina Oviedo

I (UNL) CE9 Matemática para maestros Irma Saiz (UNNE) CE l O Didáctica de la Matemática Sara Scaglia (UNL)


1 1 0 CRONICAS DE LA LJI REUNION ANUAL DE LA U . M . A .

CE l l Cómo Demostrar y Crear en Matemática. Domingo A. Tarzia (Univ. Austral-CONICET)

CE l 2 Conceptos básicos del Cálculo. Edgardo N . Guichal (Univ. Nac. del Sur)

CE13 Modelos Probabilísticos. José Martínez (Fa.MAF-UNC)

1 .2 . COMUNICACIONES CIENTIFICAS . En el . transcurso de la Reunión Anual, se llevó a cabo la "LII Reunión Anual de Comunicaciones Científicas" . Durante los días 1 9 y 20 se realizaron 1 5 sesiones en paralelo de comunicaciones habiendo sido aceptados 1 5 5 trabajos según el siguiente detalle : Anál is is Real ( 1 9) , Ecuaciones Diferenciales y Si stemas Dinámicos ( 1 8) , Geometría Diferenc ial , Teoría de Lie y Representación de Grupos (2 1 ) , Álgebras de la Lógica (24), Matemática Apl icada (24), Álgebra, Geometría Algebraica y Geometría Computacional ( 1 8) , Análisis Funcional y Análisis Numérico ( 1 4) , · Aproximaciones, Desarrollos y Funciones Especiales (5) , Teoría de Grafos, Combinatoria y Convexidad (8) . No fueron comunicados nueve trabajos aceptados .

1 . 3 . COMUNICACIONES EN EDUCACIÓN MATEMÁTICA. En la Reunión Anual, se concretó la XXV Reunión de Educación Matemática. Se expusieron durante los días 1 9 y 20 . Fueron aceptados 59 trabajos para su exposición, de los cuales no se expusieron doce.

1 .4 . CEREMONIA INAUGURAL se l l evó a cabo en el salón Paraninfo de la Universidad del Litoral .

1 .4 . 1 . Se hizo entrega del premio "Luis A n tonio Santaló " por el décimo primer concurso de monografias para estudiantes de Matemática. Tema: El teorema de GaussBonnet. Jurado : Dra. Ursula Molter, Dra. Liliana Gysin y Dr. Guillermo Keilhauer

Se adjudicó el primer premio a los alumnos María Belén Celis, José Abel Semitie l , y Natal ia Sgreccia de la Fac . de Ciencias Exactas , Ingeniería y Agrimensura de la Universiodad Nacional de Rosario y el segundo a los alumnos María Marta Cannilla y Pablo Sebastián Del ieutraz, de la Fac . de Ciencias Exactas y Naturales de la Universidad de Buenos Aires.

Durante el acto fue recordada la figura del Dr. Luis Antonio Santaló por su alumna la Dra. Graciela Birman, de la Universidad Nacional del Centro de la Pcia. de B sAs. y además fue leído un recordatorio enviado por el Dr. Norberto Fava.

1 .4 .2 . La CONFERENCIA REY PASTOR estuvo a cargo del Dr. Ricardo H. Nochetto de l a Universidad de Maryland (USA), en el tema Control del error y adaptividad para ecuaciones diferenciales en derivadas parciales.


CRONICAS DE LA LI I REUNION ANUAL DE LA O .M.A . 1 1 1

1 .4 . 3 . Posteriormente s e agasajó a l o s asistentes con un vino de honor, en l a Facultad

de Ingeniería Química.

1 . 5 . CONFERENCIAS . Además de la ya mencionada conferencia "Julio Rey Pastor" se dictaron las siguientes :

• CONFERENCIA ALBERTO GONZÁLEZ DOMíNGUEZ, a cargo del Dr. Aroldo KapIan (Universidad Nacional de Córdoba), Análisis diádico y relaciones de

conmutación.

• A lgunos problemas en la modelación matemática de dispersión líquida, a cargo

del Dr. Fabio Rosso (Universidad de Firenze, Italia)

• Sub variedades y h olonomía, a cargo del Dr. Carlos Olmos (Universidad Nacional de Córdoba)

• La investigación en Educación Matemática : qué ocurre en Argentina ? , a cargo de la Dra. Mónica Vil lareal (Universidad Nacional de Córdoba)

1 .6 . INSCRIPTOS . Registraron su inscripción al Congreso 450 asistentes.

1 .7 . AUSPICIOS Y SUBSIDIOS . L a reunión contó con subsidios de: l a Embajada de España, l a Fundación Antorchas, de CONICET y de SECyT.

Con el auspicio de las siguientes instituciones y organismos :

Universidad Nacional del Litoral Res . 1 7 1 /02 Universidad de Concepción del Uruguay 6/05/02 Universidad Nacional de Salta Res . 293/02 Universidad Nacional de la Patagonia Res . Rl5 1 5202 Universidad Nacional de Jujuy Res . R 352/02 Universidad Nacional de Entre Ríos Res. 1 94/02 Universidad Nacional de Río Cuarto Res. 403/02 Universidad Nacional de La Pampa Res .CS 1 3 1 /02 Universidad Nacional de Gral . Sarmiento Res . 330 1 02 Facultad de Ciencias Económicas , UNL Res.D/07 1 /02 Facultad Regional Santa Fe UTN Res .D/092/02 Facultad de Ingeniería , UNER Res.CD/ 1 20102 Facultad de Ciencias Económicas , UNER Res.D/046/02 Facultad de Ingeniería Química, UNL Res .CD/ 1 32/02 Facultad de Ciencia y Tecnología, UADER Res .DRl077 102 Facultad Regional Concepción del Umguav UTN Res.D/092/02 Facultad de Bioquímica}' Cs. Biológicas, UNL Res.CD/1 37/02 Facultad de Ciencias Veterinarias , UNL Res .CD/245/02 Facultad de Ciencias Agrarias , UNL Res.270/02


1 1 2 CRONICAS DE LA LIl REUNION ANUAL DE LA U.M.A.

. . Facultad de Ingeniería V Cs.Hídricas, UNL Res .CD/ 1 3 3/02 Facultad de Arquitectura, Diseño y Urbanismo,

Res .CD/057/02 UNL Facultad de Humanidades V Ciencias, UNL Res .CDI l45/02 Facultad de Ciencias Económicas , UNR Res . 330/02 Facultad Regional Paraná, UTN Res . 091 /02 Facultad de Ciencias Exactas , Ingeniería

Res .270/02 Iv Agrimensura, UNR Facultad de Economía y Administración , UN del

Res .D228/02 Comahue Ministerio de Cultura V Educación, Misiones Res .08 1 /02 Consejo Provincial de Educación, Neuquen Res .0626/02 Consejo Provincial de Educación , Rio Negro Res . 1 8 56/02 Ministerio de Educación, Salta Res 1 95/02 Ministerio de Cultura y Educación, Formosa Res 1 1 6 1 /02 Secretaría de Educación, La Rioia Res 1 204/02 Ministerio de Educación, Córdoba Res 707 Cultura y Educación, San Luis Res 50 Ministerio de Educación, Ciencia y Tecnología de Res 677 la Nación Secretaría de Educación de la Ciudad de Bs. As. Res 1 9 1 9 Ministerio de Educación Santa Fe Res 0553 Gobierno de la Pcia. de Santa Fe Dec. 1 852 Cámara de Senadores de la Pcia. de Santa Fe Decl . 0080 Cámara de Diputados de la Pcia. de Santa Fe Centro Regional de Investigación y Desarrollo- Res .0 1 /2002 CERIDE


INDICE . Volumen 44,

Número 1,2003

Weighted BMOd! spaces and the Hilbert Transform. M. Morvidone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : ............................ 1

One-Sided Singular Integral operators on Calderón-Hardy spaces. S. Ombrosi and C. Segovia . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . ... . . . . . . . . . . . . . . . . . . . . .... . . . . 17

Linear combination of a new sequence of linear positive operators. P. N. Agrawal and A. J. Mohammad . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . ... . . . . . . . . . . . . . . . . . .. . . . . .. . . . 33

On convergence of derivatives of a new sequence of linear positive operators. P. N. Agrawal and A. J. Mohammad . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . ... . . . . . .. . . . ... . . . . . . . . . . . . . . . . .43

.,

Spline wavelets in periodic Sobolev spaces and applications to high order Collocation methods. ,\; F. Bastin, C. Boigelot and P. Laubin . . . . . . . . . . . . . . . . . . : ............................ , .............................. 53

F initely generated multiresolution analysis in several variables. . A. Benavente and C. Cabrelli ....... : . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . ... . . . . . . . . . . : ........... 75

The Resolveq�Qn Rn, a degenerating method.

C. Will ... . :} .. ,.: ........................................................................... , ....... i . . . . . .. . . . . . . . . . . . . . . . . . . . .. . 89

The Adaptive dynamics for the randomly alternating Prisoner's Dilema �ame. E. EI-Sedy ..........................................

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........................................... ;': ............................ 99

Reunión Annual de la Unión Matemática Argentina . . . . . . . . . . . . . . . . . . . . . . . ! . . . . . . . . .. . . . . . . . . . . . . . , ............ 109

r;-;�� Editorial de la r�� l:Ir1iversidad Nacional del Sur UNSf7

Reg. Nac. de la Prop. In!. N° 180.863

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director de pubucaciones: maternatlea luis a. piovan i … · 2013-08-28 · 2 marcela morvidone a...

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