Approximation by Multivariate Singular Integrals

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SpringerBriefs in Mathematics

For further volumes: http://www.springer.com/series/10030

George A. Anastassiou

Approximation by Multivariate Singular Integrals

123

George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152 USA [email protected]

ISSN 2191-8198 e-ISSN 2191-8201 ISBN 978-1-4614-0588-7 e-ISBN 978-1-4614-0589-4 DOI 10.1007/978-1-4614-0589-4 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011933230 Mathematics Subject Classification (2010): 41A17, 41A25, 41A28, 41A35, 41A36, 41A60, 41A80 © George A. Anastassiou 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To my wife Koula and my daughters Angela and Peggy

Preface

This short monograph is the first to deal exclusively with the study of the approximation of multivariate singular integrals to the identity-unit operator. Here we study quantitatively the basic approximation properties of the general multivariate singular integral operators, special cases of which are the multivariate Picard, Gauss-Weierstrass, Poisson-Cauchy and trigonometric singular integral operators, etc. These operators are not general positive linear operators. In particular we study the rate of convergence of these operators to the unit operator, as well as the related simultaneous approximation. These are given via inequalities and by the use of multivariate higher order modulus of smoothness of the high order partial derivatives of the involved function. Also we study the global smoothness preservation properties of these operators. Some of these multivariate inequalities are proved to be attained, that is sharp. Furthermore we give asymptotic expansions of Voronovskaya type for the error of approximation. These properties are studied with respect to Lp norm, 1 p 1: The last chapter presents a related Korovkin type approximation theorem for functions of two variables. Plenty of examples are given. For the convenience of the reader, the chapters are self-contained. This brief monograph relies on author’s last two years of related research work, more precisely see author’s articles in the list of references of each chapter. Advanced courses can be taught out of this short book. All necessary background and motivations are given per chapter. The presented results are expected to find applications in many areas of pure and applied mathematics, such as mathematical analysis, probability, statistics and partial differential equations, etc. As such this brief monograph is suitable for researchers, graduate students, and seminars of the above subjects, also to be in all science libraries. The preparation of this book took place during 2010–2011 in Memphis, Tennessee, USA.

vii

viii

Preface

I would like to thank my family for their dedication and love to me, which was the strongest support during the writing of thismonograph. Department of Mathematical Sciences The University of Memphis, TN 38152, USA

George A. Anastassiou

Contents

1

Uniform Approximation by General Multivariate Singular Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 1 1.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 1 1.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 14 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 16

2 Lp -Approximation by General Multivariate Singular Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .

19 19 19 30 32

3 Global Smoothness Preservation and Simultaneous Approximation by Multivariate General Singular Integrals .. . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 3.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .

33 33 34 44 45 45

4 Multivariate Voronovskaya Asymptotic Expansions for General Singular Integrals . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 4.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 4.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .

47 47 47 55 56

ix

x

Contents

5 Simultaneous Approximation by Multivariate Complex General Singular Integrals .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 5.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 5.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .

57 57 57 67 68 68

6 Approximation of Functions of Two Variables via Almost Convergence of Double Sequences . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 6.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 6.2 Korovkin-Type Approximation Theorem .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 6.3 Some Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .

69 69 71 76 79

Chapter 1

Uniform Approximation by General Multivariate Singular Integral Operators

In this chapter, we present the uniform approximation properties of general multivariate singular integral operators over RN , N 1. We give their convergence to the unit operator with rates. The estimates are pointwise and uniform. The established inequalities involve the multivariate higher order modulus of smoothness. We list the multivariate Picard, Gauss-Weierstrass, Poisson Cauchy and trigonometric singular integral operators where this theory can be applied directly. This chapter relies on [2].

1.1 Introduction The rate of convergence of univariate singular integral operators has been studied in [1, 7–9, 11, 12], and these works motivate the current chapter. Here we consider some very general multivariate singular integral operators over RN , N 1, and we study the degree of approximation to the unit operator with rates over smooth functions. We present related inequalities involving the multivariate higher modulus of smoothness with respect to kk1 . The estimates are pointwise and uniform. See Theorems 1.9, 1.11. We mention particular operators that fulfill our theory. The discussed linear operators are not in general positive. Other motivation comes from [4, 5].

1.2 Main Results Here r 2 N; m 2 ZC , we define 8 r rj ˆ ˆ .1/ j m ; if j D 1; 2; : : : ; r; < j Œm ˛j;r WD r P ˆ r m ˆ :1 i ; if j D 0; .1/ri i i D1 G.A. Anastassiou, Approximation by Multivariate Singular Integrals, SpringerBriefs in Mathematics, DOI 10.1007/978-1-4614-0589-4 1, © George A. Anastassiou 2011

(1.1)

1

2

1 Uniform Approximation by General Multivariate Singular Integral Operators

and Œm

ık;r WD

r X

Œm

˛j;r j k ; k D 1; 2; : : : ; m 2 N.

(1.2)

j D1

See that

r X

Œm

˛j;r D 1;

(1.3)

r r D .1/r : j 0

(1.4)

j D0

and

r X

.1/rj

j D1

Let n be a probability Borel measure on RN , N 1, n > 0, n 2 N. We now define the multiple smooth singular integral operators Z r X Œm Œm .f I x1 ; : : : ; xN / WD ˛j;r f .x1 C s1 j; x2 C s2 j; : : : ; xN CsN j / dn .s/; r;n j D0

RN

(1.5) where s WD .s1 ; : : : ; sN /, x WD .x1 ; : : : ; xN / 2 RN I n; r 2 Z, m 2 ZC , f W RN ! R is a Borel measurable function, and also .n /n2N is a bounded sequence of positive real numbers. Œm Remark 1.1. The operators r;n are not in general positive. For example, consider PN 2 the function ' .u1 ; : : : ; uN / D i D1 ui and also take r D 2, m D 3; xi D 0, i D 1; : : : ; N . See that ' 0, however

0 Œ3 2;n

.'I 0; 0; : : : ; 0/ D @

2 X

1 Œ3 j 2 ˛j;2 A

Z

j D1

N X RN

Z Œ3 Œ3 D ˛1;2 C 4˛2;2 Z 1 D 2 C 2 RN assuming that

R RN

N P i D1

si2

dn .s/ < 1:

! si2

dn .s/

i D1 N X

RN N X i D1

! si2 dn .s/

i D1

! si2

dn .s/ < 0;

(1.6)

1.2 Main Results

3

Œm Lemma 1.2. The operators r;n preserve the constant functions in N variables.

Proof. Let f .x1 ; : : : ; xN / D c, then Œm r;n

.cI x1 ; : : : ; xn / D

r X

Œm ˛j;r

Z

j D0

Rn

cdn .s/ D c: t u

We need

Definition 1.3. Let f 2 CB RN , the space of all bounded and continuous functions on RN . Then, the rth multivariate modulus of smoothness of f is given by (see, e.g. [6]) !r .f I h/ WD p

r

sup

u1 ;u2 ;:::;uN

u21 C:::Cu2N h

.f /1 < 1;

h > 0;

(1.7)

where kk1 is the sup-norm and ru f .x/ WD ru1 ;u2 ;:::;uN f .x1 ; : : : ; xN / D

r X

.1/rj

j D0

r f .x1 C j u1 ; x2 C j u2 ; : : : ; xN C j uN / : j

(1.8)

Let m 2 N and let f 2 C m RN : Suppose that all partial derivatives of f of order m are bounded, i.e. m @ f .; ; : : : ; / @x ˛1 : : : @x ˛N < 1; 1 N 1 for all ˛j 2 ZC , j D 1; : : : ; N I

PN

j D1 ˛j

(1.9)

D m:

We make Remark 1.4.˛ Let l D 0; 1; : : : ; m. The lth order partial derivative is denoted by f˛ WD @@xf˛ , where ˛ WD .˛1 ; : : : ; ˛N /, ˛i 2 ZC , i D 1; : : : ; N and j˛j WD PN i D1 ˛i D l: Consider gz .t/ WD f .x0 C t .z x0 //, t 0; x0 ; z 2 RN . Then 2 !j 3 N X @ gz.j / .t/ D 4 .zi x0i / f 5.x01 C t .z1 x01 / ; : : : ; x0N Ct .zN x0N // ; @x i i D1 (1.10) for all j D 0; 1; : : : ; m:

4


We have the multivariate Taylor’s formula, [3]: f .z1 ; : : : ; zN / D gz .1/ D

m .j / X gz .0/

jŠ

j D0

1 C .m 1/Š

Z 0

1

.1 /m1 gz.m/ ./ gz.m/ .0/ d: (1.11)

Notice gz .0/ D f .x0 /. Also for j D 0; 1; : : : ; m, we have 0 gz.j / .0/

D

X ˛WD.˛1 ;:::;˛N /; ˛i 2ZC ; N P i D1;:::;N; j˛jWD ˛i Dj i D1

1

B B jŠ B B N BY @ ˛

i

C N ! C Y C ˛i f˛ .x0 / : .zi x0i / C C i D1 ŠA

(1.12)

i D1

Furthermore, 1

0 gz.m/ ./

D

X ˛WD.˛1 ;:::;˛N /; ˛i 2ZC ; N P i D1;:::;N; j˛jWD ˛i Dm i D1

B B mŠ B B N BY @ ˛

i

C N ! C Y C ˛i .zi x0i / f˛ .x0 C .z x0 // ; C C i D1 ŠA

i D1

(1.13) 0 1: We apply the above for z D .z1 ; : : : ; zN / D .x1 C s1 j; x2 C s2 j; : : : ; xN C sN j / D x C sj; and

x0 D .x01 ; : : : ; x0N / D .x1 ; x2 ; : : : ; xN / D x;

to obtain . jQ / m X gxCsj .0/ 1 f .x1 C s1 j; : : : ; xN C sN j / D gxCsj .1/ D C Q .m 1/Š jŠ jQD0

Z

1

0

.m/ .m/ .1 /m1 gxCsj ./ gxCsj .0/ d; (1.14)

where gxCsj .t/ WD f .x C t .sj //: Notice gxCsj .0/ D f .x/.

1.2 Main Results

5

Also for jQ D 0; 1; : : : ; m we have 1

0 X

.jQ/

gxCsj .0/ D

˛WD.˛1 ;:::;˛N /; ˛i 2ZC ; N P i D1;:::;N; j˛jWD ˛i DjQ

C N ! C Y C ˛i .si j / f˛ .x/: C C i D1 A Š

B B jQŠ B B N BY @ ˛

i

(1.15)

i D1

i D1

Furthermore, we get 1

0 .m/ gxCsj ./=mŠ

X

D

˛WD.˛1 ;:::;˛N /; ˛i 2ZC ; N P i D1;:::;N; j˛jWD ˛i Dm

B B 1 B B N BY @ ˛

i D1

i

C N ! C Y C ˛i f˛ .x C .sj // ; .si j / C C i D1 A Š

i D1

(1.16) 0 1: For P jQ D 1; : : : ; m and ˛ WD .˛1 ; : : : ; ˛N / ; ˛i 2 ZC ; i D 1; : : : ; N; Q j˛j WD N i D1 ˛i D j , will be proved that Z c˛;n;jQ WD c˛;n WD

N Y

RN

si˛i dn .s1 ; : : : ; sN / 2 R,

(1.17)

i D1

see (1.35). Consequently, we derive m X

R

jQD1

RN

.jQ/ gxCsj .0/dn .s/ jQŠ 0

D

m X jQD1

B B

B jQ B

1

0 X

j B B ˛WD.˛ ;:::;˛ /; ˛ 2ZC ; 1 N i @ N i D1;:::;N; j˛jWD

P

i D1

˛i DjQ

1

C C C C C C C c˛;n f˛ .x/C C: C C A A iŠ

B B 1 B B N BY @ ˛ i D1

(1.18)

6


Next we observe that Z Z 1 1 .m/ .m/ m1 gxCsj ./ gxCsj .0/ d dn .s/ .1 / .m 1/Š RN 0 D mj m

X

1

0

N Y

1

B C B C B C B C B ˛WD.˛1 ;:::;˛N /; ˛i 2ZC ; C B C N @ A P ˛i Dm i D1;:::;N; j˛jWD

!

˛i Š

i D1

i D1

Z

N Y

RN

Z si˛i

1

! .1 /

m1

!

Œf˛ .x C .sj // f˛ .x/ d dn .s/ :

0

i D1

(1.19) We further make Remark 1.5. We further notice that Œm r;n

.f I x/ f .x/ D

r X

Z Œm ˛j;r

RN

j D0

"

Z RN

.jQ/ m X gxCsj .0/ jQŠ

jQD1

1 C .m 1/Š

Z

.f .x C sj / f .x// dn .s/ D

r X

Œm

˛j;r

j D0 1

.1 /

m1

0

.m/ .m/ gxCsj ./gxCsj .0/

# d dn .s/: (1.20)

That is Œm Œm r;n .x/ WD r;n .f I x/ f .x/

r X

0 Œm ˛j;r @

j D0

D

1 .m 1/Š

r X j D0

Œm : DW Rr;n

Œm

˛j;r

Z

Z RN

.m/ gxCsj ./

1

0

Z

@ RN

.jQ/ m X gxCsj .0/ jQD1

jQŠ

1

1

A dn .s/A

.1 /m1

0

.m/ gxCsj .0/

! d dn .s/ (1.21)

1.2 Main Results

7

We see that 0 Œm r;n .x/

D

Œm r;n

.f I x/ f .x/

r X

Œm ˛j;r

j D1

1

B C B X c f .x/ C ˛;n ˛ C j B C N B C Y @˛1 ;:::;˛N 0W jQD1 ˛Š A m X

jQ B

j˛jDjQ

i

i D1

0

1

0 1B C m r B X c f .x/ C X X C ˛;n ˛ B Q Œm Œm j @ ˛j;r j A B D r;n .f I x/ f .x/ C N C B Y j D1 ˛ ;:::;˛ 0W Q @ 1 N j D1 ˛Š A j˛jDjQ

i

0 D

Œm r;n

.f I x/ f .x/

m X jQD1

B B

1

i D1

C c˛;n f˛ .x/ C C C: N C Y ˛ ;:::;˛ 0W @ 1 N ˛i Š A j˛jDjQ

Œm B ıjQ;r B B

X

(1.22)

i D1

So we have proved that 0 Œm r;n .x/

D

Œm r;n .f

I x/ f .x/

m X jQD1

Œm ıjQ;r

1

B C B X c f .x/ C ˛;n ˛ B C B C: N B C @˛1 ;:::;˛N 0W Y ˛ Š A j˛jDjQ

(1.23)

i

i D1

We also make Remark 1.6. We observe that 1

0 Œm Dm Rr;n

r X

Œm ˛j;r j m

j D1

Z

0 @

N Y RN

X

i D1

˛1 ;:::;˛N 0W j˛jDm

Z si˛i

1

B B 1 B B N 1BY @ ˛ A

i

C C C C C ŠA

i D1

! m1

.1 /

!

.f˛ .x C .sj // f˛ .x// d dn .s/

0

(1.24)

8


1

0 Dm

B B 1 B B N BY @ ˛

X 0

1

@

A

˛1 ;:::;˛N 0W j˛jDm

Z

N Y

RN

i

C C C C C ŠA

i D1

Z si˛i

1

.1 /

m1

0

i D1

r X

Œm

˛j;r j m

j D1

!

!

.f˛ .x C .sj // f˛ .x// d dn .s/ 0 Dm

"

B B 1 B B N BY @ ˛

X 0

1

@

A

˛1 ;:::;˛N 0W j˛jDm

r X

rj

.1/

j D1

1

Dm

˛1 ;:::;˛N 0W j˛jDm

"

r X

j D0

N Y RN

Z si˛i

1

.1 /m1 0

i D1

# ! ! r r r f˛ .x C .sj // C .1/ f˛ .x/ d dn .s/ j 0 (1.25)

1

B B 1 B B N BY @ ˛

.1/rj

Z

i D1

0 X

i

C C C C C ŠA

i

C C C C C ŠA

Z

N Y

RN

i D1

Z si˛i

1

.1 /m1

0

i D1

# ! ! r f˛ .xC .sj // d dn .s/ : j

(1.26)

We have proved that 0 Œm Rr;n

X

Dm

˛1 ;:::;˛N 0 j˛jDm

Z

N Y RN

i D1

1

B B 1 B B N BY @ ˛

i

i D1

Z si˛i

C C C C C ŠA

1 0

.1 /

m1

rs f˛ .x/

!

!

d dn .s/ :

(1.27)

1.2 Main Results

9

We further make Remark 1.7. We observe that 0 ˇ ˇ ˇ Œm ˇ .1.27/ ˇRr;n ˇ m

X ˛1 ;:::;˛N 0 j˛jDm

Z

N Y RN

1

B B 1 B B N BY @ ˛

i

i D1

Z

1

˛i

jsi j

! .1 /

X ˛1 ;:::;˛N 0 j˛jDm

Z

N Y

RN

B B 1 B B N BY @ ˛

i

Z

1

˛i

jsi j

˛1 ;:::;˛N 0 j˛jDm

RN

B B 1 B B N BY @ ˛

! .1 /

˛1 ;:::;˛N 0 j˛jDm

!

d dn .s/

i

C C C C C ŠA

i D1

Z

1

˛i

jsi j

! .1 /

0

m1

!

!r .f˛ ; ksk2 / d dn .s/

1

0

(1.28)

1

i D1

X

krs f˛ k1

dn .s/

C C C C C ŠA

0

X

N Y

m1

!

i D1

i D1

Z

jrs f˛ .x/j d

1

0 m

m1

0

i D1

0 m

C C C C C ŠA

B B 1 B B N BY @ ˛

i

C Z C C C C RN ŠA

N Y

!

!

˛i

jsi j !r .f˛ ; ksk2 / dn .s/ :

i D1

i D1

(1.29)

10


So far we have established 0 ˇ ˇ ˇ Œm ˇ ˇRr;n ˇ

B B 1 B B N BY @ ˛

X ˛1 ;:::;˛N 0 j˛jDm

i

0

˛1 ;:::;˛N 0 j˛jDm

C Z C C C C RN ŠA

i D1

!

N Y

!

˛i

jsi j !r .f˛ ; ksk2 / dn .s/

i D1

1

B B 1 B B N BY @ ˛

X

D

1

i

C Z C C C C RN ŠA

! ! ksk2 dn .s/ jsi j !r f˛ ; n n i D1 N Y

˛i

i D1

(1.30) (using the fact !r .f I u/ .1 C /r !r .f I u/ ;

(1.31)

; u > 0; we get) 1

0 X

0

1

B C @˛ ;:::;˛ 0 A 1 N

C B B ! .f ; / C Z B r ˛ n C C B N C RN B Y @ ˛i Š A

N Y

! jsi j

ksk2 1C n

˛i

i D1

!

r

dn .s/ :

i D1

j˛jDm

(1.32)

We have proved that 0 ˇ ˇ ˇ Œm ˇ ˇRr;n ˇ

X 0 B @˛

1 1 ;:::;˛N 0

C A

1

B C B ! .f ; / C Z B r ˛ n C B N C B Y C RN @ ˛i Š A

N Y

! ˛i

jsi j

i D1

ksk2 1C n

r

! dn .s/ :

i D1

j˛jDm

(1.33) We also make Remark 1.8. Notice that for j˛j D m: ! Z N Y ˛i dn .s/ jsi j RN

i D1

Z RN

N Y

! ˛i

jsi j

i D1

by the assumption in Theorem 1.9, next.

1C

(1.34) ksk2 n

r dn .s/ < 1;

1.2 Main Results

11

Hence

Z jc˛;n j

N Y

RN

for all ˛ W .˛1 ; : : : ; ˛N / W j˛j WD Hence also Z

jsi j

dn .s/ < 1;

(1.35)

i D1

P

RN

! ˛i

˛i D m, ˛i 2 ZC .

jsi jm dn .s/ < 1;

(1.36)

for all i D 1; : : : ; N: Let 1 jQ m 1, then Z

Z

jQ

RN

jsi j dn .s/

mjQ m jQ jQ dn .s/ jsi j

RN

Z

mjQ < 1: jsi j dn .s/ m

D RN

(1.37)

That is for 1 jQ m 1, we obtain that Z

Q

RN

jsi jj dn .s/ < 1:

(1.38)

Let jQ D 1; : : : ; m 1I ˛ WD .˛1 ; : : : ; ˛N /, ˛i 2 ZC , i D 1; : : : ; N; j˛j WD P ˛i D 1. Hence ˛i D jQ. That is N i D1

PN

i D1

jQ

Z

N Y RN i D1

˛i

jsi j dn .s/

N Y R i D1

D

N Y R i D1

Therefore, it holds

Z

N Y RN i D1

Based on the above, we present

RN

RN

˛i

jsi j

˛jQ

i

˛Qi j dn .s/

˛Qi j < 1: jsi j dn .s/ jQ

jsi j˛i dn .s/ < 1:

(1.39)

(1.40)

m .;:::;/ < 1, Theorem 1.9. Let m2 N, f 2 C m .RN /, N 1, x2 RN . Assume @x@ ˛f1 :::@x ˛N 1 N 1 P N for all ˛j 2 ZC , j D 1; : : : ; N W j˛j WD j D1 ˛j D m: Let n be a Borel probability measure on RN , for n > 0, .n /n2N bounded sequence.

12


C PNSuppose that for all ˛ WD .˛1 ; : : : ; ˛N / ; ˛i 2 Z , i D 1; : : : ; N; j˛j WD i D1 ˛i D m we have that

Z un WD

N Y RN

! jsi j

ksk2 1C n

˛i

i D1

r dn .s/ < 1:

(1.41)

For jQ D 1; : : : ; m, and ˛ WD .˛1 ; : : : ; ˛N /, ˛i 2 ZC , i D 1; : : : ; N; j˛j WD P N Q i D1 ˛i D j , call Z c˛;n WD c˛;n;jQ WD

N Y RN i D1

si˛i dn .s1 ; : : : ; sN / :

(1.42)

Then i) ˇ 0 1ˇ ˇ ˇ ˇ ˇ ˇ B ˇ C m ˇ B C X Œm B X c˛;n;jQ f˛ .x/ Cˇˇ ˇ Œm Œm ıjQ;r B Er;n .x/ WD ˇr;n .f I x/ f .x/ Cˇ N ˇ B Cˇ Y ˛ ;:::;˛ 0W ˇ ˇ Q @ 1 N j D1 ˛i Š A ˇ ˇ j˛jDjQ ˇ ˇ i D1

X

0

1

B C @˛ ;:::;˛ 0 A 1 N

j˛jDm

.!r .f˛ ; n // ! N Y ˛i Š

Z RN

! ! N Y ksk2 r ˛i dn .s/ ; 1C jsi j n i D1

i D1

(1.43) 8 x 2 RN : ii)

Œm Er;n

1

R:H:S: .1.43/:

(1.44)

Given that n ! 0, as n ! 1, and un is uniformly bounded, then we derive Œm that Er;n ! 0 with rates. 1 iii) It holds also that 0

Œm r;n .f / f

1

1 ˇ ˇ B C ˇ ˇ m ˇ ˇB C X ˇ Œm ˇ B X ˇc˛;n;jQˇ kf˛ k1 C ˇıjQ;r ˇ B C C R:H:S: .1.43/; N B C Y @˛1 ;:::;˛N 0W A jQD1 ˛Š j˛jDjQ

i

i D1

(1.45)

1.2 Main Results

13

given that kf˛ k1 < 1, for all ˛ W j˛j D jQ, jQ D 1; : : : ; m: Furthermore, as n ! 0 when n ! 1, assuming that c˛;n;jQ ! 0, while un is uniformly bounded, we conclude that Œm (1.46) r;n .f / f ! 0 1

with rates. Case m D 0:

Remark 1.10. Here f 2 CB RN (bounded and continuous functions). We notice that Œ0 .f I x/ f .x/ r;n Z r X Œ0 D ˛j;r .f .x C js/ f .x// dn .s/ RN

j D0

Z D

0 @

RN

Z D

D

0 @ 0 @

@ RN

@

D RN

0 ˛j;r f .x C js/ @ Œ0

r X

r X

r X

1

1

Œ0 ˛j;r A f .x/A dn .s/

(1.47)

r X

1

1

˛j;r A f .x/A dn .s/ Œ0

(1.48)

j D1

.1/rj

1 0 1 r X r r A f .x C js/ @ f .x/A dn .s/ .1/rj j j j D1

.1/rj

j D1

0

Z

r X

r X j D0

j D1

0

D

.x C js/ @

j D1

RN

Z

0 Œ0 ˛j;r f

j D0

RN

Z

r X

.1/

j D0

rj

1 r r f .x C js/ C .1/r f .x/A dn .s/ j 0 1 Z r r A s f .x/ dn .s/: f .x C js/ dn .s/ D N j R (1.49)

We established that Z Œ0 .f I x/ f .x/ D r;n

RN

r s f .x/ dn .s/:

(1.50)

14


Consequently, it holds ˇ ˇ Z ˇ Œ0 ˇ ˇr;n .f I x/ f .x/ˇ Z

RN

RN

jrs f .x/j dn .s/ Z krs f k1 dn .s/

RN

!r .f; ksk2 / dn .s/

ksk2 !r f; n dn .s/ n RN Z ksk2 r dn .s/: !r .f; n / 1C n RN Z

D

(1.51)

So we proved ˇ ˇ Z ˇ Œ0 ˇ ˇr;n .f I x/ f .x/ˇ

RN

ksk2 r dn .s/ !r .f; n / ; 1C n

(1.52)

8 x 2 R: Based on the above, we present Theorem 1.11. Let f 2 CB RN , N 1: Then Œ0 r;n f f

1

Z

(1.53)

RN

ksk2 r dn .s/ !r .f; n / ; 1C n

(1.54)

RN

ksk2 r dn .s/ < 1: 1C n

under the assumption Z ˆn WD

As n ! 1 and n ! 0, given that ˆn are uniformly bounded, we derive Œ0 r;n f f

1

!0

with rates.

1.3 Applications Let all entities as in Sect. 1.2. We define the following specific operators: (a) The general multivariate Picard singular integral operators:

(1.55)

1.3 Applications

15

Œm Pr;n .f I x1 ; : : : ; xN / WD

1 .2n /N

r X Œm ˛j;r

N P

Z RN

f .x1 Cs1 j; x2 Cs2 j; ::: ; xN CsN j /e

Observe that 1 .2n /

N P

Z N

e

(1.56)

j D0 !

jsi j

i D1 n

ds1 : : : dsN :

!

jsi j

i D1 n

ds1 : : : dsN D 1;

RN

(1.57)

see [1]. (b) The general multivariate Gauss–Weierstrass singular integral operators: Œm Wr;n .f I x1 ; : : : ; xN / WD p

1

N

n

r X

RN

f .x1 C s1 j; x2 C s2 j; : : : ; xN C sN j / e

Notice that p

N P

Z

1

N

n

e

i D1 n

(1.58)

j D0 N P

Z

Œm

˛j;r

i D1 n

! si2

ds1 : : : dsN :

! si2

RN

ds1 : : : dsN D 1;

(1.59)

see [7]. (c) The general multivariate Poisson–Cauchy singular integral operators: Œm .f I x1 ; : : : ; xN / WD WnN Ur;n

r X

Œm

˛j;r

(1.60)

j D0

Z RN

f .x1 C s1 j; : : : ; xN C sN j /

with ˛ 2 N, ˇ >

N Y i D1

1 ; 2˛

1 si2˛ C n2˛

ˇ ds1 : : : dsN ;

and 2˛ˇ1

Wn WD see [8].

.ˇ/ ˛n 1 2˛ ˇ

1 2˛

;

(1.61)

16


Observe that Z WnN

RN

N Y

1

2˛ i D1 si

C n2˛

ˇ ds1 : : : dsN D 1;

(1.62)

see [8, 13], p. 397, formula 595. (d) The general multivariate trigonometric singular integral operators: Œm Tr;n .f I x1 ; : : : ; xN / WD N n

r X

Œm

˛j;r

(1.63)

j D0

Z RN

f .x1 Cs1 j; : : : ; xN CsN j /

N Y

0 @

sin

12ˇ si n

si

i D1

A

ds1 : : : dsN ;

where ˇ 2 N, and n WD 2n12ˇ .1/ˇ ˇ

ˇ X

.1/k

kD1

k 2ˇ1 ; .ˇ k/Š .ˇ C k/Š

(1.64)

see [9, 10], p. 210, item 1033. Notice that N n

Z

N Y RN i D1

0 @

sin

12ˇ si n

si

A

ds1 : : : dsN D 1;

(1.65)

see also [9, 10], p. 210, item 1033. Œm Œm Œm Œm One can apply Theorems 1.9 and 1.11 to operators Pr;n ; Wr;n ; Ur;n , Tr;n Œm (special cases of r;n ) and derive interesting results.

References 1. G. Anastassiou, Basic Convergence with rates of smooth Picard singular integral operators, J. Comput. Anal. Appl, 8(2006), 313-334. 2. G. Anastassiou, General Uniform Approximation Theory by Multivariate Singular Integral Operators, submitted, 2011. 3. G. Anastassiou and S.S. Dragomir, On some estimates of the remainder in Taylor’s formula, J. of Math. Anal. and Appl., Vol 263, issue 1, pp. 246-263, (2001). 4. G. Anastassiou and O. Duman, Statistical Approximation by double Picard singular integral operators, Studia Univ. Babes-Bolyai Math., 55(2010), 3-20.

References

17

5. G. Anastassiou and O. Duman, Uniform Approximation in statistical sense by double Gauss-Weierstrass singular integral operators, Nonlinear Funct. Anal. Appl. (accepted for publication, 2009). 6. G. Anastassiou and S. Gal, Approximation Theory, Birkhaüser, Boston, Basel, Berlin, 2000. 7. G. Anastassiou and R. Mezei, Uniform convergence with rates of smooth Gauss-Weierstrass singular integral operators, Applicable Analysis, 88, 7(2009), 1015-1037. 8. G. Anastassiou and R. Mezei, Uniform convergence with rates of smooth Poisson-Cauchy type singular integral operators, Mathematical and Computer Modelling, 50(2009), 1553-1570. 9. G. Anastassiou and R. Mezei, Uniform convergence with rates of general singular operators, CUBO, accepted, 2011. 10. J. Edwards, A treatise on the integral calculus, Vol II, Chelsea, New York, 1954. 11. S.G. Gal, Remark on the degree of approximation of continuous functions by singular integrals, Math. Nachr., 164(1993), 197-199. 12. R.N. Mohapatra and R.S. Rodriguez, On the rate of convergence of singular integrals for Hölder continuous functions, Math. Nachr., 149(1990), 117-124. 13. D. Zwillinger, CRC standard Mathematical Tables and Formulae, 30th edition, Chapman & Hall/CRC, Boca Raton, 1995.

Chapter 2

Lp -Approximation by General Multivariate Singular Integral Operators

In this chapter, we present the Lp , 1 p < 1 approximation properties of general multivariate singular integral operators over R N , N 1. We give their convergence to the unit operator with rates. The established inequalities involve the multivariate higher order modulus of smoothness. We list the multivariate Picard, Gauss–Weierstrass, Poisson Cauchy and trigonometric singular integral operators where this theory can be applied directly. This chapter is based on [3].

2.1 Introduction The rate of Lp , 1 p < 1 convergence of univariate singular integral operators has been studied in [1, 7–9], see also the related [11, 12], and these works motivate the current chapter. Here, we consider some very general multivariate singular integral operators over RN , N 1, and we study the degree of approximation to the unit operator with rates over smooth functions. The derived related inequalities are involving the multivariate higher modulus of smoothness with respect to kkp : See Theorems 2.4, 2.6, 2.8, 2.10. We mention particular operators that fulfill this theory. The discussed linear operators are not in general positive, see [2]. Other motivation comes from [4, 5].

2.2 Main Results Here r 2 N; m 2 ZC , we define

Œm

˛j;r

8 r ˆ rj ˆ if j D 1; 2; : : : ; r; .1/ j m ; ˆ < j r WD X r m ˆ ri ˆ 1 i ; if j D 0; .1/ ˆ : i

(2.1)

i D1

G.A. Anastassiou, Approximation by Multivariate Singular Integrals, SpringerBriefs in Mathematics, DOI 10.1007/978-1-4614-0589-4 2, © George A. Anastassiou 2011

19

2 Lp -Approximation by General Multivariate Singular Integral Operators

20

and Œm ık;r

WD

r X

Œm

˛j;r j k ; k D 1; 2; : : : ; m 2 N.

(2.2)

j D1

See that r X

Œm

˛j;r D 1;

(2.3)

r r r D .1/ : j 0

(2.4)

j D0

and

r X

.1/

rj

j D1

Let n be a probability Borel measure on RN , N 1, n > 0, n 2 N. We now define the multiple smooth singular integral operators Œm r;n

.f I x1 ; : : : ; xN / WD

r X

Z Œm ˛j;r

j D0

RN

f .x1 C s1 j; x2 C s2 j; : : : ; xN C sN j /dn .s/ ; (2.5)

where s WD .s1 ; : : : ; sN /, x WD .x1 ; : : : ; xN / 2 RN I n; r 2 N, m 2 ZC , f W RN ! R is a Borel measurable function, and also .n /n2N is a bounded sequence of positive real numbers. Œm The operators r;n preserve constants, see [2]. Here, we deal with f 2 C m RN , m 2 ZC , with f˛ 2 Lp RN , j˛j D m 2 ZC , p 1I where f˛ denotes the mixed partial P Q Q j˛j WD N j D1 ˛j D j , j D 1; : : : ; m: We need

@jQ f .;:::;/ ˛ ˛ @x1 1 :::@xNN

, ˛j 2 ZC , j D 1; : : : ; N W

Definition 2.1 (see also [6]). We call ru f .x/ WD ru1 ;u2 ;:::;uN f .x1 ; : : : ; xN / (2.6) r X r WD .1/rj f .x1 C j u1 ; x2 C j u2 ; : : : ; xN C j uN / : j j D0

Let p 1, the modulus of smoothness of order r is given by !r .f I h/p WD sup ru .f /p ; kuk2 h

h > 0.

(2.7)

2.2 Main Results

21

I) First, we consider the case of m 2 N, p > 1: We make RemarkP 2.2. For jQ D 1; : : : ; m, and ˛ WD .˛1 ; : : : ; ˛N /, ˛i 2 ZC , i D 1; : : : ; N; Q j˛j WD N i D1 ˛i D j , under the assumption of Theorem 2.4, Z c˛;n;jQ WD c˛;n WD

N Y RN i D1

si˛i dn .s1 ; : : : ; sN / 2 R;

(2.8)

see also [2]. From [2], we obtain 1

0 Œm Er;n

.x/ WD

Œm r;n

.f I x/ f .x/

m X jQD1

0

C B B X c f .x/ C ˛;n ˛ C C N C Y @j˛jDjQ ˛i Š A

Œm B ıjQ;r B B

i D1

1

B X B B 1 Dm B N BY j˛jDm @ ˛

i

(2.9)

C Z C C C C RN ŠA

N Y

Z si˛i

i D1

1

.1 /m1 0

i D1

rs f˛

!

!

.x/ d dn .s/

Œm .x/ ; 8 x 2 RN : DW Rr;n

Let p; q > 1 W

1 p

C

1 q

D 1. Then

ˇ ˇp ˇ ˇp ˇ Œm ˇ ˇ Œm ˇ .x/ˇ ˇEr;n .x/ˇ D ˇRr;n 0 0 0

11p 1

B B B B X B 1 B B B B Bset c1 WD Bm B N B B B @ j˛jDm @ Y ˛ @ ˇZ ˇ ˇ D c1 ˇ ˇ RN Z c1

(2.10)

C C C C C A

i D1 N Y

Z si˛i

i D1 N Y

RN

i

CC CC CC CC CC Š AA

i D1

1

.1 /

0

Z jsi j

m1

˛i

!

1

.1 / 0

ˇp ! ˇ r ˇ s f˛ .x/ d dn .s/ˇ ˇ

m1

jrs f˛

!p

.x/j d dn .s/

: (2.11)


22

Hence, we have Z ˇ ˇp ˇ ˇ Œm I1 WD ˇEr;n .x/ˇ dx

(2.12)

RN

Z

Z c1

RN

N Y RN

Z jsi j

1

˛i

! .1 /

m1

0

i D1

Call 0 .s; x/ WD

Z

N Y

jsi j

.x/j d dn .s/

.1 /

jrs f˛

Z

Z RN

!p dx

! m1

0

i D1

D c1

1

˛i

jrs f˛

.x/j d

p RN

dx DW I2 :

.s; x/ dn .s/

(2.13)

Therefore, it holds Z

Z I 2 c1

.s; x/ dn .s/ dx DW I3 : p

RN

RN

(2.14)

But we have

.s; x/

N Y

Z jsi j

D

q q1 Z m1 d .1 /

0

i D1 N Y

1

˛i

jsi j

0

Z

1

˛i

1

.q .m 1/ C 1/ q

i D1

1

jrs f˛

1 0

jrs f˛

p

.x/j d

p

p1

.x/j d

(2.15)

p1 :

Hence, we obtain p .s; x/

N Y

jsi j˛i p

i D1

Z

1

1

p

.q .m 1/ C 1/ q

0

jrs f˛ .x/jp d :

(2.16)

Thus, we get Z I3 c2

RN

Z

N Y

RN i D1

Z

1

jsi j˛i p 0

!

jrs f˛ .x/jp d dn .s/ dx

(2.17)

2.2 Main Results

23

set c2 WD c1 Z D c2

Z D c2

p

.q .m 1/ C 1/ q

Z

RN

!

1 Z

1

RN i D1

0 N Y

RN i D1

N Y

Z jsi j

1

˛i p

! jsi j

˛i p

.x/j dx d dn .s/

jrs f˛

.x/jp dx d dn .s/ DW I4 : (2.18)

Z RN

0

!

p

jrs f˛

Consequently, we derive Z I4 c2

N Y RN i D1

Z c2

RN

p !r .f˛ I ksk2 /p d dn .s/

! jsi j˛i p !r .f˛ I ksk2 /pp dn .s/

i D1 N Y

RN

1

0

N Y

Z D c2

Z jsi j˛i p

! ˛i p

jsi j

i D1

ksk2 p !r f˛ I n dn .s/ n p

(2.19)

by !r .f; h/p .1 C /r !r .f; h/p ; for any h; > 0; p 1 ! ! rp Z N Y ksk 2 c2 !r .f˛ I n /pp dn .s/ : 1C jsi j˛i p n RN i D1

(2.20)

We have proved that 20 Z RN

0

3p

11

6B B ˇp 6B X B 1 ˇ ˇ 6B B ˇ Œm B N ˇEr;n .x/ˇ dx 6B 6B B 4@j˛jDm @ Y ˛

i


!7 7 7 7 1 q .q .m 1/ C 1/ 7 5 m

i D1

"

Z !r .f˛ I n /pp

RN

! ! #p ksk2 r dn .s/ : 1C jsi j n i D1 N Y

˛i

(2.21)


24

Thus, we derive (p > 1) 11

0

0

B !B BX B 1 B B B N B 1 B B q .q .m 1/ C 1/ @j˛jDm @ Y ˛

Œm Er;n .x/

m

p

i


i D1

"Z

"

RN

!

N Y

ksk2 1C n

˛i

jsi j

i D1

# p1

r #p

dn .s/

!r .f˛ I n /p :

(2.22)

We make Remark 2.3. Notice that (p > 1) Z

N Y

RN

"Z RN

! jsi j

ksk2 1C n

˛i

i D1

N Y

!

ksk2 1C n

jsi j˛i

i D1

r dn .s/

(2.23)

# p1

r !p dn .s/

< 1,

by assumption of Theorem 2.4. As in [2] then we get that Z

N Y

RN

jsi j˛i dn .s/ < 1:

(2.24)

i D1

Hence, c˛;n;jQ 2 R. From the above, we have proved Theorem 2.4. Let f 2 C m RN , m 2 N, N 1, with f˛ 2 Lp RN ; j˛j D m, x 2 RN : Let p; q > 1 W p1 C q1 D 1. Here, n is a Borel probability measure on RN C for n > 0, .n /n2N bounded PN sequence. Assume for all ˛ WD .˛1 ; : : : ; ˛N / ; ˛i 2 Z , i D 1; : : : ; N; j˛j WD i D1 ˛i D m that we have Z

N Y RN

i D1

! jsi j

˛i

ksk2 1C n

r !p dn .s/ < 1:

(2.25)

2.2 Main Results

25

For jQ D 1; : : : ; m; and ˛ WD .˛1 ; : : : ; ˛N / ; ˛i 2 ZC , i D 1; : : : ; N; j˛j WD P N Q i D1 ˛i D j , call Z Y N c˛;n;jQ WD si˛i dn .s/ : (2.26) RN i D1

Then 0 1 B C m B X c Qf˛ .x/ C X ˛;n;j Œm Œm C Œm B ıjQ;r B Er;n D r;n .f I x/ f .x/ ! C N B C p @j˛jDjQ Y ˛ Š A jQD1 i 1

0

!B BX 1 B B 1 N B Y q .q .m 1/ C 1/ @j˛jDm ˛ m

i

i D1

(2.27) p;x

C C C C C ŠA

i D1

"Z

" RN

N Y

!

ksk2 1C n

˛i

jsi j

i D1

# p1

r #p

dn .s/

!r .f˛ ; n /p :

Œm ! 0 with rates. As n ! 1 and n ! 0, by (2.27), we obtain that Er;n p One also finds by (2.27) that Œm r;n .f I x/ f .x/ 0

1 ˇ ˇ ˇ ˇ ˇc˛;n;jQ ˇ

m ˇ ˇB X ˇ Œm ˇ B X ˇıjQ;r ˇ B N Y @

jQD1

j˛jDjQ

˛i Š

p;x

C C kf˛ kp C C R:H:S: .2.27/; A

i D1

given that kf˛ kp < 1; j˛j D jQ, jQ D 1; : : : ; m: Œm .f / f p ! 0, Assuming that c˛;n;jQ ! 0, n ! 0; as n ! 1, we get r;n Œm that is r;n ! I the unit operator, in Lp norm, with rates.

II) Case of m D 0, p > 1. We make Remark 2.5. In [2] we proved that Z Œ0 .f r;n

I x/ f .x/ D RN

rs f .x/ dn .s/ :

(2.28)


26

Let p; q > 1 W

1 p

C

1 q

D 1. Hence

ˇ ˇp Z ˇ Œ0 ˇ .f I x/ f .x/ ˇr;n ˇ Z

RN

RN

jrs f

p .x/j dn .s/

(2.29)

jrs f .x/jp dn .s/ :

And it holds Z RN

ˇ ˇp ˇ Œ0 ˇ ˇr;n .f I x/ f .x/ˇ dx Z

Z

RN

Z

RN

Z

RN

.x/j dn .s/ dx p

Z

D RN

jrs f

(2.30)

RN

jrs f

p

.x/j dx dn .s/

(2.31)

!r .f; ksk2 /pp dn .s/

ksk2 p D !r f; n dn .s/ n p RN Z ksk2 rp p dn .s/ : 1C !r .f; n /p n RN Z

(2.32)

Therefore, we obtain Œ0 r;n .f / f Z RN

We proved

p

(2.33)

p1 ksk2 rp dn .s/ !r .f; n /p : 1C n

Theorem 2.6. Let f 2 C RN \ Lp RN ; N 1; p; q > 1 W p1 C q1 D 1. Assume n probability Borel measures on RN , .n /n2N > 0 and bounded. Also suppose Z ksk2 rp 1C dn .s/ < 1: (2.34) n RN

2.2 Main Results

27

Then Œ0 r;n .f / f Z RN

(2.35)

p

p1 ksk2 rp dn .s/ !r .f; n /p : 1C n

Œ0 Œ0 .f / f p ! 0, i.e. r;n ! I , the unit As n ! 0, when n ! 1, we derive r;n operator, in Lp norm. III) Next follows the case m D 0, p D 1. We make Remark 2.7. As before we have ˇ ˇ Z ˇ Œ0 ˇ ˇr;n .f I x/ f .x/ˇ

RN

jrs f .x/j dn .s/ :

(2.36)

Hence Z Œ0 r;n .f / f D 1

RN

ˇ ˇ ˇ ˇ Œ0 ˇr;n .f I x/ f .x/ˇ dx Z

Z RN

RN

jrs f .x/j dn .s/ dx

Z

Z

(2.38)

D RN

(2.37)

RN

jrs f

.x/j dx dn .s/

Z RN

Z D RN

!r .f; ksk2 /1 dn .s/ ksk2 !r f; n dn .s/ n 1 Z

!r .f; n /1

RN

(2.39)

ksk2 r dn .s/ : 1C n

We have proved

Theorem 2.8. Let f 2 C RN \ L1 RN , N 1: Assume n probability Borel measures on RN , .n /n2N > 0 and bounded. Also suppose Z RN

ksk2 r 1C dn .s/ < 1: n

(2.40)


28

Then Œ0 r;n .f / f

(2.41)

1

ksk2 r dn .s/ !r .f; n /1 : 1C n

Z RN

Œ0 As n ! 0, we get r;n ! I in L1 norm.

IV) Case of m 2 N, p D 1. We make Remark 2.9. We have 0 Z Œm Er;n D 1

RN

1

B ˇ ˇ X B ˇ B 1 ˇ Œm B N ˇEr;n .x/ˇ dx m BY j˛jDm @ ˛

i

C C C C C ŠA

i D1

Z

Z RN

N Y

RN

Z jsi j

!

1

˛i

.1 /

m1

0

i D1

jrs f˛

!

.x/j d dn .s/ dx (2.42)

0

0

11

B B B X B B B 1 Bset c1 WD m B N B BY j˛jDm @ @ ˛

i

CC CC CC CC CC ŠA A

i D1

Z D c1

N Y RN i D1

Z c1

Z jsi j

N Y RN

1

˛i

.1 /

RN

0

! Z jsi j

Z m1

1

˛i 0

i D1

jrs f˛

.x/j dx d dn .s/

.1 /m1 !r .f˛ ; ksk2 /1 d dn .s/ (2.43)

c1 m

Z

N Y RN

i D1

! jsi j

˛i

!r .f˛ ; ksk2 /1 dn .s/

2.2 Main Results

29

1

0

CZ C C C C RN ŠA

B X B B 1 D B N BY j˛jDm @ ˛

i

0

i D1

B X B B 1 B N BY j˛jDm @ ˛

N Y

! ˛i

jsi j

i D1

ksk2 dn .s/ !r f˛ ; n n 1

(2.44)

1

i

C Z C C C !r .f˛ ; n /1 C RN ŠA

N Y

! jsi j

˛i

1C

i D1

ksk2 n

r dn .s/ :

i D1

(2.45) We have proved

Theorem 2.10. Let f 2 C m RN , m; N 2 N, with f˛ 2 L1 RN ; j˛j D m, x 2 RN : Here, n is a Borel probability measure on RN for n > 0, .n /n2N is C a bounded P sequence. Suppose for all ˛ WD .˛1 ; : : : ; ˛N / ; ˛i 2 Z , i D 1; : : : ; N; ˛ D m that we have j˛j WD N i D1 i Z

N Y

RN

i D1

! ˛i

jsi j

ksk2 1C n

r ! dn .s/ < 1:

(2.46)

C Q For PN j D 1; : : : ; m; and ˛ WD .˛1 ; : : : ; ˛N / ; ˛i 2 Z , i D 1; : : : ; N; j˛j WD Q i D1 ˛i D j , call Z Y N c˛;n;jQ WD si˛i dn .s/ : (2.47) RN i D1

Then 0 1 B C m B C X Œm B X c˛;n;jQ f˛ .x/ C Œm Œm ıjQ;r B Er;n D r;n .f I x/ f .x/ C N B C 1 Y @j˛jDjQ jQD1 ˛i Š A i D1 1;x 0 1 B X B B 1 B N BY j˛jDm @ ˛

i

C Z C C C !r .f˛ ; n /1 C RN ŠA

i D1

Œm ! 0 with rates. As n ! 0, we get Er;n 1

N Y i D1

! jsi j

˛i

ksk2 1C n

(2.48)

r dn .s/ :


30

From (2.48) we get Œm r;n f f

1

0

1 ˇ ˇ B C ˇ ˇ m ˇ ˇB C X ˇ Œm ˇ B X ˇc˛;n;jQ ˇ C kf ˇıjQ;r ˇ B ˛ k1 C C R:H:S: .2:48/; N B C @j˛jDjQ Y ˛ Š A jQD1 i i D1

given that kf˛ k1 < 1; j˛j D jQ, jQ D 1; : : : ; m: Œm As n ! 1, assuming n ! 0 and c˛;n;jQ ! 0, we obtain r;n f f 1 ! 0, Œm that is r;n ! I in L1 norm, with rates.

2.3 Applications Let all entities as in Sect. 2.2. We define the following specific operators: (a) The general multivariate Picard singular integral operators: Œm .f I x1 ; : : : ; xN / WD Pr;n

1

r X

.2n /N

j D0

Œm

˛j;r

N P

Z RN


Notice that 1 .2n /

N P

Z N

e RN

(2.49)

!

jsi j

i D1 n

ds1 : : : dsN :

!

jsi j

i D1 n

ds1 : : : dsN D 1;

(2.50)

see [1]. (b) The general multivariate Gauss–Weierstrass singular integral operators: r X 1 Œm Œm .f I x1 ; : : : ; xN / WD p ˛j;r Wr;n N j D0 n N P

Z RN


i D1 n

! si2

ds1 : : : dsN :

(2.51)

2.3 Applications

31

Observe that p

1

N

n

!

N P

Z

e

i D1 n

si2

ds1 : : : dsN D 1;

RN

(2.52)

see [7]. (c) The general multivariate Poisson–Cauchy singular integral operators: Œm Ur;n .f I x1 ; : : : ; xN / WD WnN

r X

Œm

˛j;r

(2.53)

j D0

Z RN

f .x1 C s1 j; : : : ; xN C sN j /

N Y i D1

with ˛ 2 N, ˇ >

1 ; 2˛

1 si2˛ C n2˛

ˇ ds1 : : : dsN ;

and 2˛ˇ1

Wn WD

.ˇ/ ˛n 1

2˛

ˇ

;

(2.54)

ˇ ds1 : : : dsN D 1;

(2.55)

1 2˛

see [8]. Notice that Z WnN

N Y

RN i D1

1 si2˛ C n2˛

see [8, 13], p. 397, formula 595. (d) The general multivariate trigonometric singular integral operators: Œm Tr;n .f I x1 ; : : : ; xN / WD N n

r X

Œm

˛j;r

(2.56)

j D0

Z RN

f .x1 C s1 j; : : : ; xN C sN j /

N Y

0 @

i D1

sin

12ˇ si n

si

A

ds1 : : : dsN ;


ˇ X kD1

see [9, 10], p. 210, item 1033.

.1/k

k 2ˇ1 ; .ˇ k/Š .ˇ C k/Š

(2.57)


32

Observe that N n

Z

N Y RN i D1

0 @

sin

12ˇ si n

si

A

ds1 : : : dsN D 1;

(2.58)

see also [9, 10], p. 210, item 1033. Œm Œm One can apply Theorems 2.4, 2.6, 2.8, and 2.10 to operators Pr;n ; Wr;n ; Œm Œm Œm Ur;n , Tr;n (special cases of r;n ) and derive interesting results.

References 1. G. Anastassiou, Lp convergence with rates of smooth Picard singular operators, Differential and difference equations and applications, Hindawi Publ. Corp., New York, (2006), 31-45. 2. G. Anastassiou, General uniform Approximation Theory by Multivariate singular integral operators, submitted, 2011. 3. G. Anastassiou, Lp -general Approximations by Multivariate Singular Integral Operators, submitted, 2011. 4. G. Anastassiou and O. Duman, Statistical Lp -approximation by double Gauss-Weierstrass singular integral operators, Comput. Math. Appl. 59(2010), 1985-1999. 5. G. Anastassiou and O. Duman, Statistical Lp -convergence of double smooth Picard singular integral operators, J. Comput. Anal. Appl., Vol. 13, No. 1(2011), 167-187. 6. G. Anastassiou and S. Gal, Approximation Theory, Birkhaüser, Boston, Basel, Berlin, 2000. 7. G. Anastassiou and R. Mezei, Lp convergence with rates of smooth Gauss-Weierstrass singular operators, Nonlinear Studies, accepted, 2011. 8. G. Anastassiou and R. Mezei, Lp convergence with rates of smooth Poisson-Cauchy type singular operators, International Journal of Mathematical Sciences, accepted, 2010. 9. G. Anastassiou and R. Mezei, Lp convergence with rates of general singular integral operators, submitted, 2010. 10. J. Edwards, A treatise on the integral calculus, Vol II, Chelsea, New York, 1954. 11. S.G. Gal, Remark on the degree of approximation of continuous functions by singular integrals, Math. Nachr., 164(1993), 197-199. 12. R.N. Mohapatra and R.S. Rodriguez, On the rate of convergence of singular integrals for Hölder continuous functions, Math. Nachr., 149(1990), 117-124. 13. D. Zwillinger, CRC standard Mathematical Tables and Formulae, 30th edition, Chapman & Hall/CRC, Boca Raton, 1995.

Chapter 3

Global Smoothness Preservation and Simultaneous Approximation by Multivariate General Singular Integrals

In this chapter, we continue with the study of multivariate smooth general singular integral operators over RN , N 1, regarding their simultaneous global smoothness preservation property with respect to the Lp norm, 1 p 1, by involving multivariate higher order moduli of smoothness. Also, we present their multivariate simultaneous approximation to the unit operator with rates. The derived multivariate Jackson-type inequalities are almost sharp containing elegant constants, and they reflect the high order of differentiability of the engaged function. In the uniform case of global smoothness we prove optimality. At the end we list the multivariate Picard, Gauss–Weierstrass, Poisson–Cauchy and Trigonometric singular integral operators as applicators of this general theory. This chapter relies on [4].

3.1 Introduction The main motivation for this chapter comes from [1, 5, 6]. We give here the multivariate simultaneous global smoothness preservation property of multivariate general smooth singular integral operators. We study also the simultaneous Lp , 1 p 1, approximation of these operators to the unit operator with rates. See Theorems 3.2, 3.4, Proposition 3.5, Theorem 3.8, Corollary 3.9 and Theorems 3.10–3.15. At the end, we list specific operators that fulfill our theory. One can find many interesting convergence properties based on these results.


33

34

3

Global Smoothness and Simultaneous Approximation by Singular Integrals


Œm

˛j;r

8 r ˆ rj ˆ j m ; if j D 1; 2; : : : ; r; ˆ < .1/ j r WD X r m ˆ ˆ ˆ .1/ri i ; if j D 0; :1 i

(3.1)

i D1

and Œm

ık;r WD

r X

Œm

˛j;r j k ; k D 1; 2; : : : ; m 2 N.

(3.2)

j D1

See that r X

Œm

˛j;r D 1;

(3.3)

r r D .1/r : j 0

(3.4)

j D0

and

r X

.1/rj

j D1

Let n be a probability Borel measure on RN , N 1, n > 0, n 2 N. We now define the multiple smooth singular integral operators Œm .f r;n

I x1 ; : : : ; xN / WD

r X j D0

Z Œm ˛j;r

RN

f .x1 C s1 j; x2 C s2 j; : : : ; xN C sN j / dn .s/ ;

(3.5)

where s WD .s1 ; : : : ; sN /, x WD .x1 ; : : : ; xN / 2 RN I n; r 2 N, m 2 ZC , f W RN ! R is a Borel measurable function, and also .n /n2N is a bounded sequence of positive real numbers. Œm Above operators r;n are not in general positive operators and they preserve constants, see [2]. Definition 3.1. Let f 2 C RN , N 1, m 2 N, the mth modulus of smoothness for 1 p 1, is given by !m .f I h/p WD sup km t .f /kp;x ; kt k2 h

(3.6)

3.2 Main Results

35

h > 0, where m t f

.x/ WD

m X

.1/

mj

j D0

m f .x C jt/ : j

(3.7)

Denote !m .f I h/1 D !m .f; h/ :

(3.8)

Above, x; t 2 RN : We present the general global smoothness preservation result

Œm Q .f I x/ 2 R, 8 x 2 R. Let h > 0, f 2 C RN , Theorem 3.2. We assume r;n N 1. i) Assume !m .f; h/ < 1. Then

0 1 r ˇ ˇ X ˇ ˇ Œ m Q Œm !m r;nQ f; h @ ˇ˛jQ;r ˇA !m .f; h/ :

(3.9)

jQD0

ii) Assume f 2 C RN \ L1 RN : Then 0 1 r ˇ ˇ X ˇ ˇ Œ m Q Œm Q f; h @ !m r;n ˇ˛jQ;r ˇA !m .f; h/1 :

(3.10)

iii) Assume f 2 C RN \ Lp RN , p > 1: Then 0 1 r ˇ ˇ X ˇ Œ mQ ˇA Œm Q !m r;n f; h @ ˇ˛jQ;r ˇ !m .f; h/p :

(3.11)

1

p

jQD0

jQD0

Proof. We recall (x 2 RN , N 1) Œm Q r;n .f

I x/ D

r X jQD0

Œ Q ˛jQm ;r

Z RN

f x C s jQ dn .s/ :

We see (t 2 RN ) m X m Œm mj Œm Q Q .f I x/ D .1/ .f I x C jt/ r;n m t r;n j j D0

D

m X j D0

.1/mj

(3.12)

1 0 Z r X m Œ Q @ ˛jQm f x C jt C jQk A dn.k/ ;r j RN jQD0

(3.13)

36

3


D

r X jQD0

D

r X jQD0

Q ˛jQm ;r

Q ˛jQm ;r

Œ

Œ

0

Z

@ RN

Z

m X

.1/mj

j D0

RN

1 m f xCjtCjQk A dn .k/ j

Q m t f x C j k dn .k/ :

(3.14)

That is we find m t

Z r X Œm Q Œm Q ˛jQ;r r;n .f I x/ D jQD0

RN

m t f x C jQk dn .k/ :

(3.15)

Therefore r ˇ ˇ ˇZ ˇ X ˇ ˇ m Œ mQ ˇ Œ mQ ˇ ˇt r;n .f I x/ ˇ ˇ˛jQ;r ˇ jQD0

RN

ˇ m ˇ ˇ f x C jQk ˇ d .k/ : n t

(3.16)

(a) From the last (3.16), we derive r ˇ ˇZ X ˇ Œ mQ ˇ Œm Q !m r;n f; h ˇ˛jQ;r ˇ jQD0

RN

!m .f; h/ dn .k/

(3.17)

0

1 r ˇ ˇ X ˇ ˇ Œm Q D@ ˇ˛jQ;r ˇA !m .f; h/ ; jQD0

proving the claim (3.9). (b) We see that Z RN

r ˇ ˇ ˇZ ˇ X ˇ m Œ mQ ˇ Œ mQ ˇ ˇ ˇt r;n .f I x/ ˇ dx ˇ˛jQ;r ˇ jQD0

Z RN

RN

ˇ m ˇ ˇ f x C jQk ˇ dx t

dn .k/ D

r ˇ ˇZ X ˇ Œ mQ ˇ ˇ˛jQ;r ˇ

jQD0

RN

km t f k1 dn .k/

0

1 r ˇ ˇ X ˇ Œ mQ ˇA m D@ ˇ˛jQ;r ˇ kt f k1 : jQD0

(3.18)

3.2 Main Results

37

So we derive 0 1 r ˇ ˇ X m Œ mQ ˇ Œm Q ˇ t r;n .f / @ ˇ˛jQ;r ˇA km t f k1 ;

(3.19)

0 1 r ˇ ˇ X ˇ Œ mQ ˇA Œm Q f; h @ !m r;n ˇ˛jQ;r ˇ !m .f; h/1 ;

(3.20)

1

jQD0

which implies

1

jQD0

proving the claim (3.10). (c) Let p; q > 1 W p1 C q1 D 1. Then m Œ mQ t r;n .f I x/

p;x

r ˇ ˇ Z X ˇ Œ mQ ˇ ˇ˛jQ;r ˇ jQD0

RN

ˇ m ˇ ˇ f x C jQk ˇ d .k/ n t

p;x

(3.21) D

r ˇ ˇ Z X ˇ Œ mQ ˇ ˇ˛jQ;r ˇ

Z

RN

jQD0

ˇ m ˇ ˇ f x C jQk ˇ t

RN

p dn .k/ r ˇ ˇ Z X ˇ Œ mQ ˇ ˇ˛jQ;r ˇ jQD0

Z

RN

RN

p1 dx

(3.22)

ˇ m ˇ ˇ f x C jQk ˇp t

p1 dn .k/ dx r ˇ ˇ Z X ˇ Œ mQ ˇ D ˇ˛jQ;r ˇ jQD0

RN

Z RN

dn .k/ r ˇ ˇ Z X ˇ Œ mQ ˇ D ˇ˛jQ;r ˇ jQD0

RN

ˇ m ˇ ˇ f x C jQk ˇp dx t

p1

km t f

kpp

1 r ˇ ˇ X ˇ ˇ Œ m Q D@ ˇ˛jQ;r ˇA km t f kp :

p1 dn .k/

0

jQD0

(3.23)

38

3


0 1 r ˇ ˇ X m Œ mQ ˇ Œm Q ˇ t r;n .f / @ ˇ˛jQ;r ˇA km t f kp :

That is

p

t u

Now clearly (3.24) proves (3.11). Œm ˛0;1

Remark 3.3. Let r D 1, then 1;nmQ .f I x/ D Œ

D 0,

(3.24)

jQD0

Œm ˛1;1

D 1. Hence

Z RN

f .x C s/ dn .s/ DW n .f I x/ :

(3.25)

By Theorem 3.2, we get

Theorem 3.4. We suppose n .f I x/ 2 R, 8 x 2 R. Let h > 0, f 2 C RN , N 1. i) Assume !m .f; h/ < 1. Then !m .n f; h/ !m .f; h/ :

(3.26)

ii) Assume f 2 C RN \ L1 RN : Then !m .n f; h/1 !m .f; h/1 :

(3.27)

iii) Assume f 2 C RN \ Lp RN , p > 1: Then !m .n f; h/p !m .f; h/p :

(3.28)

Next, we get an optimality result Proposition 3.5. Above inequality (3.26): !m .n f; h/ !m .f; h/ is sharp, namely it is attained by any fj .x/ D xjm ; j D 1; : : : ; N; x D x1 ; : : : ; xj ; : : : ; xN 2 RN :

(3.29)

Proof. We observe that (for x; t 2 RN ) m t fj .x/ D

m X i D0

.1/mi

m m xj C i tj i

D mŠtjm ; j D 1; : : : ; N:

(3.30)

3.2 Main Results

39

ˇ ˇm ˇ ˇm !m fj ; h D sup mŠ ˇtj ˇ D mŠ sup ˇtj ˇ D mŠhm :

So that

kt k2 h

(3.31)

kt k2 h

Also m t

Z n fj I x D

RN

f .x C s/ dn .s/ m t j

(3.32)

Z D

RN

mŠtjm dn .s/ D mŠtjm , j D 1; : : : ; N; etc, t u

proving the claim. We need

Theorem 3.6. Let f 2 C l RN , l; N 2 N. Here, n is a Borel probability measure on RN ; n > 0, .n /n2N P a bounded sequence. Let ˇ WD .ˇ1 ; : : : ; ˇN /, N N ˇi 2 ZC , i D 1; : : : ; N I jˇj WD i D1 ˇi D l: Here f .x C sj /, x; s 2 R , is n -integrable wrt s, for j D 1; : : : ; r: There exist n -integrable functions hi1 ;j ; hˇ1 ;i2 ;j ; hˇ1 ;ˇ2 ;i3 ;j ; : : : ; hˇ1 ;ˇ2 ;:::;ˇN 1 ;iN ;j 0 (j D 1; : : : ; r) on RN such that ˇ ˇ ˇ @i1 f .x C sj / ˇ ˇ ˇ ˇ hi1 ;j .s/ ; i1 D 1; : : : ; ˇ1 ; ˇ i1 ˇ ˇ @x1

(3.33)

ˇ ˇ ˇ @ˇ1 Ci2 f .x C sj / ˇ ˇ ˇ ˇ hˇ1 ;i2 ;j .s/ ; i2 D 1; : : : ; ˇ2 ; ˇ ˇ1 i2 ˇ ˇ @x @x 2

1

:: :

ˇ ˇ ˇ @ˇ1 Cˇ2 C:::CˇN 1 CiN f .x C sj / ˇ ˇ ˇ ˇ ˇ hˇ1 ;ˇ2 ;:::;ˇN 1 ;iN ;j .s/ ; iN D 1; : : : ; ˇN ; ˇ @x iN @x ˇN 1 : : : @x ˇ2 @x ˇ1 ˇ N

N 1

2

1

8x; s 2 RN . Then, both of the next exist and Œm Œm Q Q .f I x/ D r;n r;n fˇ I x : ˇ

Proof. By H. Bauer [7], pp. 103–104.

(3.34) t u

Corollary 3.7. (to Theorem 3.6, r D 1) It holds .n .f I x//ˇ D n fˇ I x : We present simultaneous global smoothness results.

(3.35)

40

3


Theorem 3.8. Let h > 0 and assumptions of Theorem 3.6 are valid. Here D 0; ˇ, (0 D .0; : : : ; 0/). i) Assume !m f ; h < 1. Then 0 1 r ˇ ˇ X ˇ ˇ Œ m Q Œm !m r;nQ .f / ; h @ ˇ˛jQ;r ˇA !m f ; h :

(3.36)

jQD0

ii) Additionally suppose f 2 L1 RN : Then !m

Œm Q .f / r;n

0 1 r ˇ ˇ X ˇ Œ mQ ˇA ;h @ ˇ˛jQ;r ˇ !m f ; h 1 : 1

(3.37)

jQD0

iii) Additionally suppose f 2 Lp RN , p > 1: Then 0 1 r ˇ ˇ X ˇ ˇ Œm Q Œm Q .f / ; h @ !m r;n ˇ˛jQ;r ˇA !m f ; h p :

p

(3.38)

jQD0

It follows Corollary 3.9. Let h > 0 and assumptions of Corollary 3.7 are valid. Here D 0; ˇ. i) Assume !m f ; h < 1. Then !m .n .f // ; h !m f ; h :

(3.39)

ii) Additionally assume f 2 L1 RN : Then !m .n .f // ; h 1 !m f ; h 1 :

(3.40)

iii) Additionally assume f 2 Lp RN , p > 1: Then !m .n .f // ; h p !m f ; h p :

(3.41)

Next comes multi-simultaneous approximation. Theorem 3.10. Let f 2 C mCl RN , m; l;N 2 N. The assumptions of Theorem 3.6 are valid. Call D 0; ˇ. Assume f C˛ 1 < 1 and Z

N Y RN

i D1

! jsi j

˛i

1C

ksk2 n

r dn .s/ < 1;

3.2 Main Results

41

PN for all ˛j 2 ZC , j D 1; : : : ; N; j˛j WD j D1 ˛j D m, where n is a Borel N probability measure on R , for n > 0, .n /n2N bounded sequence. C Q PNFor j D 1; : : : ; m; and ˛ WD .˛1 ; : : : ; ˛N / ; ˛i 2 Z , i D 1; : : : ; N; j˛j WD Q i D1 ˛i D j , call Z Y N c˛;n;jQ WD si˛i dn .s/ : (3.42) RN i D1

Then 1 0 C B m C B X Œm B X c˛;n;jQ f C˛ ./ C Œm ıjQ;r B (3.43) C r;n .f I / f ./ N C B Y ˛ ;:::;˛ 0W Q A @ 1 N j D1 ˛i Š j˛jDjQ i D1 1 ! ! Z r N Y ˛ X !r f C˛ ; n ksk2 i dn .s/ : 1C jsi j ! N n 1 0 Y RN i D1 ˛i Š A @ ˛1 ;:::;˛N 0 j˛jDm

i D1

Proof. Based on Theorems 6 and 9 of [2]. t u N l Theorem 3.11. Let f 2 CB R , l; N 2 N (functions l-times continuously differentiable and bounded). The assumptions of Theorem 3.6 are valid. Call D 0; ˇ. Suppose Z ksk2 r dn .s/ < 1: (3.44) 1C n RN Then Œ0 f f r;n

Z RN

1

ksk2 r 1C dn .s/ !r f ; n : n

(3.45) t u

Proof. By Theorems 6 and 11 of [2]. We continue with

Theorem 3.12. Let f 2 C mCl RN ; m; l; N 2 N: The assumptions of Theorem 3.6 are valid. Call D 0; ˇ. Let f. C˛/ 2 Lp RN , j˛j D m, x 2 RN , and p; q > 1 W p1 C q1 D 1: Here, n is a Borel probability measure on RN for n > 0, .n /n2N bounded Assume for all ˛ WD .˛1 ; : : : ; ˛N /, ˛i 2 ZC , i D 1; : : : ; N; Psequence. N j˛j WD i D1 ˛i D m we have that Z RN

N Y i D1

! jsi j

˛i

ksk2 1C n

r !p dn .s/ < 1:

(3.46)

42

3


C Q For PN j D 1; : : : ; m, and ˛ WD .˛1 ; : : : ; ˛N /, ˛i 2 Z , i D 1; : : : ; N; j˛j WD Q i D1 ˛i D j , call Z Y N c˛;n;jQ WD si˛i dn .s/ : (3.47) RN i D1

Then 0 1 B C m B X c Q f C˛ .x/ C X ˛;n;j Œm C Œm B ıjQ;r B r;n .f I x/ f .x/ C N B C Y @j˛jDjQ A jQD1 ˛i Š 1

0

!B BX B B 1 B q .q .m 1/ C 1/ @j˛jDm m

i D1

p;x

C C C ! C N C Y ˛i Š A 1

(3.48)

i D1

"

"Z RN

N Y

! jsi j

ksk2 1C n

˛i

i D1

r #p

# p1 dn .s/

!r f C˛ ; n p :

Proof. By Theorems 6 and 4 of [3].

t u

We give also

Theorem 3.13. Let f 2 C l RN ; l; N 2 N: The assumptions of Theorem 3.6 are valid. Call D 0; ˇ. Let f 2 Lp RN , x 2 RN ; p; q > 1 W p1 C q1 D 1: Assume n probability Borel measures on RN ; .n /n2N > 0 and bounded. Also suppose Z RN

ksk2 rp 1C dn .s/ < 1: n

Then Z Œ0 f f r;n

p

RN

p1 ksk2 rp dn .s/ !r f ; n p : 1C n

(3.49)

Proof. By Theorems 6 and 6 of [3]. t u Theorem 3.14. Let f 2 C l RN ; l; N 2 N: The assumptions of Theorem 3.6 are valid. Call D 0; ˇ. Let f 2 L1 RN , x 2 RN : Assume n probability Borel measures on RN ; .n /n2N > 0 and bounded. Also assume

3.2 Main Results

43

Z RN

ksk2 r dn .s/ < 1: 1C n

Then Z Œ0 .f / f r;n

1

RN

ksk2 r dn .s/ !r f ; n 1 : 1C n

(3.50)

Proof. By Theorems 6 and 8 of [3]. t u Theorem 3.15. Let f 2 C mCl RN , m; l; N 2 N. The assumptions of Theorem 3.6 are valid. Call D 0; ˇ. Let f. C˛/ 2 L1 RN ; j˛j D m, x 2 RN . Here, n is a Borel probability measure on RN for n > 0, .n /n2N is a bounded P sequence. Suppose for all ˛ WD .˛1 ; : : : ; ˛N /, ˛i 2 ZC , i D 1; : : : ; N; j˛j WD N i D1 ˛i D m; we have that ! r ! Z N Y ksk2 ˛i (3.51) dn .s/ < 1: 1C jsi j n RN i D1 C Q For PN j D 1; : : : ; m, and ˛ WD .˛1 ; : : : ; ˛N /, ˛i 2 Z , i D 1; : : : ; N; j˛j WD Q i D1 ˛i D j , call Z Y N c˛;n;jQ WD si˛i dn .s/ : (3.52) RN i D1

Then 0 1 B C m B X c Q f. C˛/ .x/ C X ˛;n;j Œm C Œm B ıjQ;r B r;n .f I x/ f .x/ C N B C Y @j˛jDjQ A jQD1 ˛i Š i D1

1

0 B XB B 1 B N BY j˛jDm @ ˛

i

C Z C C C !r f C˛ ; n 1 C RN ŠA

N Y

! jsi j

i D1

˛i

1C

(3.53) 1;x

ksk2 n

r dn .s/ :

i D1

Proof. Based on Theorem 6 and Theorem 10 of [3].

t u

44

3



r X

1

.2n /N j D0

Œm

˛j;r

N P

Z RN


(3.54)

!

jsi j

i D1 n

ds1 : : : dsN :

(b) The general multivariate Gauss–Weierstrass singular integral operators: Œm .f Wr;n

1

I x1 ; : : : ; xN / WD p N n

r X

RN


(3.55)

j D0 N P

Z

Œm

˛j;r

i D1 n

! si2

ds1 : : : dsN :

(c) The general multivariate Poisson–Cauchy singular integral operators: Œm Ur;n .f I x1 ; : : : ; xN / WD WnN

r X

Œm

˛j;r

(3.56)

j D0

Z RN

f .x1 C s1 j; : : : ; xN C sN j /

N Y i D1

with ˛ 2 N, ˇ >

1 ; 2˛

1 si2˛ C n2˛

ˇ ds1 : : : dsN ;

and 2˛ˇ1

Wn WD

.ˇ/ ˛n 1 2˛ ˇ

1 2˛

:

(3.57)

(d) The general multivariate trigonometric singular integral operators: Œm .f I x1 ; : : : ; xN / WD N Tr;n n

r X

Œm

˛j;r

(3.58)

j D0

Z RN

f .x1 C s1 j; : : : ; xN C sN j /

N Y i D1

0 @

sin

12ˇ si n

si

A

ds1 : : : dsN ;

References

45

where ˇ 2 N, and

n WD 2n12ˇ .1/ˇ ˇ

ˇ X kD1

.1/k

k 2ˇ1 : .ˇ k/Š .ˇ C k/Š

(3.59)

Œm Œm Œm Œm One can apply the results of this chapter to the operators Pr;n ; Wr;n ; Ur;n , Tr;n Œm (special cases of r;n ) and obtain interesting results.

3.4 Conclusion The approximation results here imply important convergence properties of operators Œm r;n to the unit operator.

References 1. G. Anastassiou, Global smoothness and uniform convergence of smooth Picard singular operators, Comput. Math. Appl. 50(2005), no. 10-12, 1755-1766. 2. G. Anastassiou, General uniform Approximation theory by multivariate singular integral operators, submitted, 2011. 3. G. Anastassiou, Lp -general approximations by multivariate singular integral operators, submitted, 2011. 4. G. Anastassiou, Global Smoothness Preservation and Simultaneous Approximation for Multivariate General Singular Integral Operators, Appl. Math. Letters, 24(2011), 1009-1016. 5. G. Anastassiou, R. Mezei, Global smoothness and uniform convergence of smooth GaussWeierstrass singular operators, Mathematical and Computer Modelling, 50(2009), 984-998. 6. G. Anastassiou, R. Mezei, Global smoothness and uniform convergence of smooth Poisson-Cauchy type singular operators, Applied Mathematics and Computation, 215(2009), 1718-1731. 7. H. Bauer, M˛ˇ-und Integrations theorie, de Gruyter, Berlin, 1990.

Chapter 4

Multivariate Voronovskaya Asymptotic Expansions for General Singular Integrals

In this chapter, we continue with the study of approximation properties of smooth general singular integral operators over RN , N 1. We present multivariate Voronovskaya asymptotic type results and give quantitative results regarding the rate of convergence of multivariate singular integral operators to unit operator. We list specific multivariate singular integral operators that fulfill this theory. This chapter is based on [2].

4.1 Introduction The main motivation for this chapter comes from [3–5]. We give here multivariate Voronovskaya-type asymptotic expansions regarding the multivariate singular integral operators, see Theorem 4.2 and Corollaries 4.3, 4.4. In Theorem 4.6, we present the simultaneous corresponding Voronovskaya asymptotic expansion for these operators. The expansions give also the rate of convergence of multivariate general singular integral operators to unit operator. In Sect. 4.3, we list the multivariate singular Picard, Gauss–Weierstrass, Poisson–Cauchy and Trigonometric operators that fulfill the main results.


Œm

˛j;r

8 r ˆ rj ˆ j m ; if j D 1; 2; : : : ; r; ˆ < .1/ j r WD X r m ˆ ˆ ˆ .1/ri i ; if j D 0; :1 i

(4.1)

i D1


47

48

4

Voronovskaya Asymptotic Expansions for General Singular Integrals

and Œm ık;r

WD

r X

Œm

˛j;r j k ; k D 1; 2; : : : ; m 2 N.

(4.2)

j D1

Observe that r X

Œm

˛j;r D 1;

(4.3)

r r D .1/r : j 0

(4.4)

j D0

and

r X

.1/rj

j D1

Let n be a probability Borel measure on RN , N 1, n > 0, n 2 N. We now define the multiple smooth singular integral operators Œm r;n

.f I x1 ; : : : ; xN / WD

r X j D0

Z Œm ˛j;r

RN

f .x1 C s1 j; x2 C s2 j; : : : ; xN C sN j /

dn .s/ ;

(4.5)

where s WD .s1 ; : : : ; sN /, x WD .x1 ; : : : ; xN / 2 RN I n; r 2 N, m 2 ZC , f W RN ! R is a Borel measurable function, and also .n /n2N is a bounded sequence of positive real numbers. Œm The above r;n are not in general positive operators and they preserve constants, see [1]. We make Remark 4.1. Here f 2 C m RN , m; N 2 N. Let l D 0; 1; : : : ; m. The lth order ˛ partial derivative is denoted by f˛ WD @@xf˛ , where ˛ WD .˛1 ; : : : ; ˛N /, ˛i 2 ZC , P i D 1; : : : ; N and j˛j WD N i D1 ˛i D l: Consider gz .t/ WD f .x0 C t .z x0 //, t 0; x0 ; z 2 RN . Then 2 !j 3 N X @ .zi x0i / f 5 .x01 C t .z1 x01 / ; : : : ; x0N gz.j / .t/ D 4 @xi i D1 C t.zN x0N // ; for all j D 0; 1; : : : ; m: In particular, we choose z D .z1 ; : : : ; zN / D .x1 C s1 j; x2 C s2 j; : : : ; xN C sN j / D x C sj;

(4.6)

4.2 Main Results

49

and x0 D .x01 ; : : : ; x0N / D .x1 ; x2 ; : : : ; xN / D x; to get gxCsj .t/ WD f .x C t .sj // : Notice gxCsj .0/ D f .x/ : Also for jQ D 0; 1; : : : ; m 1 we get 1

0 X

.jQ/

gxCsj .0/ D

˛WD.˛1 ;:::;˛N /; ˛i 2ZC ; i D1;:::;N; j˛jWD

N X

B B jQŠ B B N BY @ ˛

˛i DjQ

i

C N ! C Y C ˛i .si j / f˛ .x/ : C C i D1 A Š

(4.7)

i D1

i D1

Furthermore, we obtain 1

0

mŠ

B B 1 B B N BY @ ˛

X

.m/

gxCsj ./

D

˛WD.˛1 ;:::;˛N /; ˛i 2ZC ; i D1;:::;N; j˛jWD

N X

i

˛i Dm

C N ! C Y C ˛i .si j / f˛ .x C .sj // ; C C i D1 ŠA

i D1

i D1

(4.8) 0 1: For P jQ D 1; : : : ; m 1; and ˛ WD .˛1 ; : : : ; ˛N / ; ˛i 2 ZC ; i D 1; : : : ; N; Q j˛j WD N i D1 ˛i D j , we define Z c˛;n;jQ WD c˛;n WD

N Y RN i D1

si˛i dn .s1 ; : : : ; sN / :

(4.9)

Consequently, we derive m X jQD1

R RN

.jQ/ gxCsj .0/ dn .s/ jQŠ 0

B B B m1 X QB j B j B D B B Q j D1 B @

0 X ˛WD.˛1 ;:::;˛N /; ˛i 2ZC ; i D1;:::;N; j˛jWD

N X i D1

˛i DjQ

1

B B 1 B B N BY @ ˛ i D1

1

C C C C C C C C C c˛;n f˛ .x/C : C C C A C iŠ A

(4.10)

50

4


Next, we observe by multivariate Taylor’s formula that f .x C js/ D gxCjs .1/ D

m1 X jQD0

. jQ / .m/ gxCjs .0/ gxCjs ./ ; C mŠ jQŠ

(4.11)

where 2 .0; 1/, which leads to Z RN

.f .x C sj / f .x// dn .s/

(4.12) 1

0

C B C BX 1 C D c j B f .x/ ! C ˛ Q ˛;n;j N C B Y A @j˛jDjQ jQD1 ˛i Š m1 X

jQ B

i D1

0

B X B B 1 Cj B N BY j˛jDm @ ˛ m

i

1

CZ N C Y C si˛i f˛ .x C sj / dn .s/ : C C RN i D1 ŠA

i D1

Hence Œm r;n .f I x/ f .x/ D

r X

Œm

˛j;r

j D0

D

Z RN

.f .x C sj / f .x// dn .s/ 1

0

r m1 XX

B BX

Œm Q B ˛j;r j j B

jQD1 j D1

B @j˛jDjQ

i D1

0 C

r X j D1

Œm ˛j;r j m

C C C f˛ .x/C c ! Q ˛;n; j N C Y A ˛i Š 1

B X B B 1 B N BY j˛jDm @ ˛

i

i D1

f˛ .x C sj / dn .s/

1

CZ N C Y C si˛i C C RN i D1 ŠA

(4.13)

4.2 Main Results

51

1

0 m1 X

D

C B B X c Q f˛ .x/ C ˛;n;j C B ! C B N C B Y @j˛jDjQ ˛Š A

Œm ıjQ;r

jQD1

(4.14)

i

i D1

1

0 B X B B 1 C B N BY j˛jDm @ ˛

i

CZ C C C C RN ŠA

N Y

! si˛i

i D1

i D1

1 0 r X r .1/rj f˛ .x C sj /A dn .s/ : @ j j D1

Thus, we have 1

0

C B B X c Q f˛ .x/ C ˛;n; j C B Œm Œm WD r;n .f I x/ f .x/ ıjQ;r B ! C N C B @j˛jDjQ Y ˛ Š A jQD1 m1 X

(4.15)

i

i D1

1

0 B X B B 1 D B N BY j˛jDm @ ˛

i

CZ C C C C RN ŠA

1 !0 r X r si˛i @ .1/rj f˛ .x C sj /A dn .s/ : j

N Y i D1

j D1

i D1

(4.16) Call ˛ .x; s/ WD

r X

.1/rj

j D1

Thus,

r f˛ .x C sj / : j

(4.17)

1

0 B X B B 1 D B N BY j˛jDm @ ˛

i

i D1

CZ C C C C RN ŠA

N Y i D1

! si˛i

˛ .x; s/ dn .s/ :

(4.18)

52

4


Consider n WD

nm

:

(4.19)

Suppose f˛ is bounded for all ˛ W j˛j D m; by M > 0. That is kf˛ k1 M . Therefore, 1 0 r X r A M D .2r 1/ M: (4.20) j˛ .x; s/j @ j j D1

Consequently, 0 r ˇ ˇ ˇn ˇ .2 1/ M nm

0

1

1

B B BX B 1 B B B B N B B @j˛jDm @ Y ˛

i

CZ C C C C RN ŠA

N Y

! jsi j˛i

i D1

C C C dn .s/C : C A

(4.21)

i D1

Suppose for j˛j D m that nm

Z RN

N Y

! jsi j

˛i

dn .s/ ; for any .n /n2N :

(4.22)

i D1

Therefore, 0

1

B C BX C ˇ ˇ 1 B C ˇ ˇ .2r 1/ M B ! C DW , > 0: n N B C @j˛jDm Y ˛ Š A i

(4.23)

i D1

Hence

j j n j j and n ! 0, nm nm where 0 < 1, as n ! 0C. That is j j m ! 0, as n ! 0C; n m : which means D 0 n

(4.24)

(4.25)

4.2 Main Results

53

We established

Theorem 4.2. Let f 2 C m RN , m; N 2 N,with all kf˛ k1 M; M > 0, all ˛ W j˛j D m. Let n > 0, .n /n2N bounded sequence, n probability Borel measures on RN . R QN ˛i dn .s/ ; all j˛j D jQ D 1; : : : ; m 1: Suppose Call c˛;n;jQ D RN i D1 si Q R N ˛i dn .s/ , all ˛ W j˛j D m; > 0, for any such .n /n2N : nm RN i D1 jsi j Also 0 < 1, x 2 RN : Then 1

0 Œm .f I x/ f .x/ D r;n

m1 X jQD1

Œm ıjQ;r

C B B X c Q f˛ .x/ C ˛;n;j C B ! C C 0 nm : B N C B @j˛jDjQ Y ˛ Š A

(4.26)

i

i D1

When m D 1, the sum collapses. Œm Above we assume r;n .f I x/ 2 R, 8 x 2 RN : @f Corollary 4.3. Let f 2 C 1 RN , N 1,with all @x M; M > 0, i D i 1 1; : : : ; N . Let n > 0, .n /n2N bounded sequence, n probability Borel measures on RN . Suppose n1

Z RN

jsi j dn .s/ ; all i D 1; : : : ; N;

(4.27)

> 0, for any such .n /n2N : Also 0 < 1, x 2 RN : Then Œ1 r;n .f I x/ f .x/ D 0 n1 : Œ1 .f I x/ 2 R, 8 x 2 RN : Above we assume r;n 2 Corollary 4.4. Let f 2 C 2 R2 , with all @@xf2 1

1

(4.28)

2 2 f ; @@xf2 ; @x@1 @x 2 2

1

1

M; M

> 0. Let n > 0, .n /n2N bounded sequence, n probability Borel measures on R2 . Call Z Z c1 D s1 dn .s/ ; c2 D s2 dn .s/ : (4.29) R2

R2

Suppose n2

Z R2

s12 dn

.s/ ;

n2

Z R2

s22 dn

.s/ ;

n2

Z R2

js1 j js2 j dn .s/ ;

54

4


> 0, for any such .n /n2N : Also 0 < 1, x 2 R2 : Then 0 1 r X @f @f Œ2 Œ2 r;n .f I x/ f .x/ D @ ˛j;r j A c1 .x/ C c2 .x/ C 0 n2 : (4.30) @x1 @x2 j D1 We continue with

Theorem 4.5. Let f 2 C l RN , l; N 2 N. Here, n is a Borel probability measure on RN ; n > 0, .n /n2N P a bounded sequence. Let ˇ WD .ˇ1 ; : : : ; ˇN /, N N ˇi 2 ZC , i D 1; : : : ; N I jˇj WD i D1 ˇi D l: Here f .x C sj /, x; s 2 R , is n -integrable wrt s, for j D 1; : : : ; r: There exist n -integrable functions hi1 ;j ; hˇ1 ;i2 ;j ; hˇ1 ;ˇ2 ;i3 ;j ; : : : ; hˇ1 ;ˇ2 ;:::;ˇN 1 ;iN ;j 0 (j D 1; : : : ; r) on RN such that ˇ ˇ ˇ @i1 f .x C sj / ˇ ˇ ˇ ˇ ˇ hi1 ;j .s/ ; i1 D 1; : : : ; ˇ1 ; i1 ˇ ˇ @x1 ˇ ˇ ˇ @ˇ1 Ci2 f .x C sj / ˇ ˇ ˇ ˇ hˇ1 ;i2 ;j .s/ ; i2 D 1; : : : ; ˇ2 ; ˇ ˇ ˇ ˇ @x i2 @x 1 2

1

:: :

ˇ ˇ ˇ @ˇ1 Cˇ2 C:::CˇN 1 CiN f .x C sj / ˇ ˇ ˇ ˇ ˇ hˇ1 ;ˇ2 ;:::;ˇN 1 ;iN ;j .s/ ; iN D 1; : : : ; ˇN ; ˇ @x iN @x ˇN 1 : : : @x ˇ2 @x ˇ1 ˇ N

N 1

2

1

(4.31) 8 x; s 2 R . Then, both of the next exist and N

Œm Œm fˇ I x : .f I x/ D r;n r;n ˇ

Proof. By H. Bauer [6], pp. 103–104. We finish this chapter with

(4.32) t u

Theorem 4.6. Let f 2 C mCl RN , m; l; N 2 N. Assumptions of Theorem 4.5 are valid. Call D 0; ˇ. Suppose f C˛ 1 M; M > 0, for all ˛ W j˛j D m. Let n > 0, .n /n2N bounded sequence, n probability Borel measures on RN . R Q N ˛i Call c˛;n;jQ D RN dn .s/ ; all j˛j D jQ D 1; : : : ; m 1: Assume i D1 si R QN ˛i dn .s/ , all ˛ W j˛j D m; > 0, for any such .n /n2N : nm RN i D1 jsi j Also 0 < 1, x 2 RN : Then

4.3 Applications

55

1

0

C B m1 C X Œm B B X c˛;n;jQ f C˛ .x/ C Œm ıjQ;r B r;n .f I x/ f .x/ D C C 0 nm : (4.33) ! N C B

Y A @j˛jDjQ jQD1 ˛Š i

i D1

When m D 1, the sum collapses.


1

r X

.2n /N

j D0

Œm

˛j;r

0

1

B B @

jsi jC A

N X

Z RN

(4.34)

C

i D1 n


ds1 : : : dsN :

(b) The general multivariate Gauss–Weierstrass singular integral operators: Œm .f I x1 ; : : : ; xN / WD p Wr;n

1

N

r X

RN

(4.35)

j D0

n

0

1

B B @

C si2 C A

N X

Z

Œm

˛j;r


i D1 n

ds1 : : : dsN :

(c) The general multivariate Poisson–Cauchy singular integral operators: Œm .f I x1 ; : : : ; xN / WD WnN Ur;n

r X

Œm

˛j;r

j D0

Z RN

f .x1 C s1 j; : : : ; xN C sN j /

N Y

2˛ i D1 si

1 C n2˛

ˇ ds1 : : : dsN ;

(4.36)

56

4

with ˛ 2 N, ˇ >


1 ; 2˛

and 2˛ˇ1

Wn WD

.ˇ/ ˛n 1 ˇ 2˛

1 2˛

:

(4.37)

(d) The general multivariate trigonometric singular integral operators: Œm Tr;n .f I x1 ; : : : ; xN / WD N n

r X

Œm

˛j;r

(4.38)

j D0

Z RN

f .x1 C s1 j; : : : ; xN C sN j /

N Y

0 @

i D1

sin

12ˇ si n

si

A

ds1 : : : dsN ;


ˇ X kD1

.1/k

k 2ˇ1 : .ˇ k/Š .ˇ C k/Š

(4.39)

Œm Œm Œm Œm ; Wr;n ; Ur;n , Tr;n One can apply the results of this chapter to the operators Pr;n Œm (special cases of r;n ) and obtain interesting results.

References 1. G. Anastassiou, General uniform Approximation theory by multivariate singular integral operators, submitted, 2011. 2. G. Anastassiou, Multivariate Voronovskaya asymptotic expansions for general singular operators, submitted, 2011. 3. G. Anastassiou, R. Mezei, A Voronovskaya type theorem for Poisson-Cauchy type singular operators, J. Math. Anal. Appl. 366(2010), 525-529. 4. G. Anastassiou, R. Mezei, Voronovskaya type theorems for smooth Picard and GaussWeierstrass singular operators, Studia Mathematica, Babes-Bolyai, Vol. LV, No. 4 (2010), 175-184. 5. G. Anastassiou, R. Mezei, A Voronovskaya type theorem for general singular operators with applications, Nonlinear Functional Analysis & Appl., accepted, 2011. 6. H. Bauer, M˛ˇ-und Integrations theorie, de Gruyter, Berlin, 1990.

Chapter 5

Simultaneous Approximation by Multivariate Complex General Singular Integrals

Here, we present complex multivariate simultaneous approximation for general smooth singular integral operators converging with rates to the unit operator. The associated and presented inequalities are in kkp , 1 p 1 norm and they involve multivariate related moduli of smoothness. At the end, we list as this theory’s applicators the special cases of multivariate complex Picard, Gauss–Weierstrass, Poisson–Cauchy and Trigonometric singular integral operators. This chapter relies on [2].

5.1 Introduction Here, we are motivated by [1, 3, 4] and expand these works to complex valued functions. We present simultaneous approximation in kkp , 1 p 1, of multivariate general smooth singular integral operators to the unit operator with rates. At the end, we list specific operators where our theory can be applied. From our approximation results, one can derive interesting convergence properties of these general operators. The expansion to complex case is based on basic properties of complex numbers and complex valued functions.


Œm

˛j;r

8 r ˆ rj ˆ j m ; .1/ if j D 1; 2; : : : ; r; ˆ < j r WD X r m ˆ ri ˆ i ; if j D 0; .1/ 1 ˆ : i

(5.1)

i D1


57

58

5 Simultaneous Approximation by Multivariate Complex General Singular Integrals

and Œm

ık;r WD

r X

Œm

˛j;r j k ; k D 1; 2; : : : ; m 2 N.

(5.2)

j D1

See that

r X

Œm

˛j;r D 1;

(5.3)

r r D .1/r : j 0

(5.4)

j D0

and

r X

.1/rj

j D1

Let n be a probability Borel measure on RN , N 1, n > 0, n 2 N. We now define the real multiple smooth singular integral operators Œm .f I x1 ; : : : ; xN / WD r;n

r X

Œm

˛j;r

j D0

Z RN

f .x1 C s1 j; x2 C s2 j; : : : ; xN C sN j / dn .s/ ;

(5.5)

where s WD .s1 ; : : : ; sN /, x WD .x1 ; : : : ; xN / 2 RN I n; r 2 N, m 2 ZC , f W RN ! R is a Borel measurable function, and also .n /n2N is a bounded sequence of positive real numbers. Œm Above operators r;n are not in general positive operators and they preserve constants, see [3]. Definition 5.1. Let f 2 C RN , N 1; m 2 N, the mth modulus of smoothness for 1 p 1, is given by !m .f I h/p WD sup km t f .x/kp;x ;

(5.6)

kt k2 h

h > 0, where m t f

.x/ WD

m X j D0

.1/

mj

m f .x C jt/ : j

(5.7)

Denote !m .f I h/1 D !m .f; h/ :

(5.8)

Above, x; t 2 RN : We make Remark 5.2. We consider here complex valued p Borel measurable functions f W RN ! C such that f D f1 C if2 , i D 1, where f1 ; f2 W RN ! R are implied to be real valued Borel measurable functions.

5.2 Main Results

59

We define the complex singular operators Œm Œm Œm .f I x/ WD r;n .f1 I x/ C ir;n .f2 I x/ ; x 2 RN : r;n

Œm We assume that r;n fj I x 2 R, 8 x 2 RN , j D 1; 2: One notices easily that ˇ ˇ ˇ Œm ˇ ˇr;n .f I x/ f .x/ˇ ˇ ˇ ˇ ˇ ˇ ˇ Œm ˇ ˇ Œm .f1 I x/ f1 .x/ˇ C ˇr;n .f2 I x/ f2 .x/ˇ ˇr;n

(5.9)

(5.10)

also Œm r;n .f I x/ f .x/ 1;x Œm Œm r;n .f1 I x/ f1 .x/ C r;n .f2 I x/ f2 .x/ 1;x

(5.11) 1;x

and Œm r;n .f / f p Œm Œm r;n .f1 / f1 C r;n .f2 / f2 ; p 1: p

(5.12)

p

Furthermore, it holds f˛ .x/ D f1;˛ .x/ C if2;˛ .x/ ;

(5.13)

where ˛ denotes a partial derivative of any order and arrangement. Here based on Theorem 9 of [3], we obtain Theorem 5.3. Let f W RN ! C, N 1, such thatf D f1 C if2 , j D 1; 2. Here m 2 N, fj 2 C m RN , x 2 RN : Assume fj;˛ 1 < 1, for all ˛k 2 ZC , P k D 1; : : : ; N W j˛j D N kD1 ˛k D mI j D 1; 2: Let n be a Borel probability measure on RN , for n > 0, .n /n2N bounded sequence. PN Assume that for all ˛ WD .˛1 ; : : : ; ˛N / ; ˛k 2 ZC , k D 1; : : : ; N; j˛j WD kD1 ˛k D m we have that ! Z N Y ksk2 r ˛k 1C dn .s/ < 1: (5.14) jsk j n RN kD1

C Q For PN j D 1; : : : ; m, and ˛ WD .˛1 ; : : : ; ˛N /, ˛k 2 Z , k D 1; : : : ; N; j˛j WD Q kD1 ˛k D j , call

Z c˛;n;jQ WD

N Y RN

kD1

sk˛k dn .s1 ; : : : ; sN / :

(5.15)

60


Then

0 1 B C m B C X Œm B X c˛;n;jQ f˛ .x/ C Œm ıjQ;r B r;n .f I x/ f .x/ C N B C Y @˛1 ;:::;˛N 0W jQD1 ˛k Š A j˛jDjQ kD1

X

1;x

.!r .f1;˛ ; n / C !r .f2;˛ ; n // ! N Y ˛k Š A 1

0 @ ˛1 ;:::;˛N 0 j˛jDm

Z RN

(5.16)

kD1

N Y

! jsk j

kD1

˛k

ksk2 1C n

!

r

dn .s/ ; x 2 RN :

The m D 0 case follows Corollary 5.4. Let f W RN ! C W f D f1 C if2 , N 1. Here j D 1; 2. Let fj 2 CB RN (continuous and bounded functions). Then Z ksk2 r Œ0 1C dn .s/ (5.17) r;n f f 1 n RN .!r .f1 ; n / C !r .f2 ; n // ; by assuming

Z RN

ksk2 r dn .s/ < 1: 1C n

(5.18)

t u Theorem 5.5 ([1]). Let f 2 C l RN , l; N 2 N. Here, n is a Borel probability measure on RN ; n > 0, .n /n2NP a bounded sequence. Let ˇ WD .ˇ1 ; : : : ; ˇN /, N ˇi 2 ZC , i D 1; : : : ; N I jˇj WD N i D1 ˇi D l: Here, f .x C sj /, x; s 2 R , is n -integrable wrt s, for j D 1; : : : ; r: There exist n -integrable functions hi1 ;j ; hˇ1 ;i2 ;j ; hˇ1 ;ˇ2 ;i3 ;j ; : : : ; hˇ1 ;ˇ2 ;:::;ˇN 1 ;iN ;j 0 (j D 1; : : : ; r) on RN such that ˇ ˇ ˇ @i1 f .x C sj / ˇ ˇ ˇ (5.19) ˇ hi1 ;j .s/ ; i1 D 1; : : : ; ˇ1 ; ˇ i1 ˇ ˇ @x1 ˇ ˇ ˇ @ˇ1 Ci2 f .x C sj / ˇ ˇ ˇ ˇ ˇ hˇ1 ;i2 ;j .s/ ; i2 D 1; : : : ; ˇ2 ; ˇ1 i2 ˇ ˇ @x2 @x1 :: : Proof. By Theorem 11 of [3].

5.2 Main Results

61

ˇ ˇ ˇ @ˇ1 Cˇ2 C:::CˇN 1 CiN f .x C sj / ˇ ˇ ˇ ˇ ˇ hˇ1 ;ˇ2 ;:::;ˇN 1 ;iN ;j .s/ ; iN D 1; : : : ; ˇN ; ˇ @x iN @x ˇN 1 : : : @x ˇ2 @x ˇ1 ˇ N 1

N

2

1

(5.1) 8 x; s 2 R . Then, bothof the next exist and N

Œm Œm Q Q .f I x/ D r;n r;n fˇ I x :

(5.20)

ˇ

We give N Theorem 5.6. NLet f W R ! C such that f D f1 C if2 : Here j D 1; 2. Let mCl R , m; l;N 2 N: For fj , the assumptions of Theorem 5.5 are valid. fj 2 C Call D 0; ˇ: Assume f. C˛/ 1 < 1 and

Z

N Y

RN

! 1C

˛k

jsk j

kD1

ksk2 n

r dn .s/ < 1;

(5.21)

where n is a Borel probability measure on RN , for n > 0, .n /n2N is bounded P sequence; for all ˛k 2 ZC , k D 1; : : : ; N W j˛j D N kD1 ˛k D m: Q D 1; : : : ; m, and ˛ WD .˛1 ; : : : ; ˛N /, ˛k 2 ZC , k D 1; : : : ; N; j˛j WD For j PN Q kD1 ˛k D j , call Z Y N sk˛k dn .s/ : c˛;n;jQ WD RN kD1

Then 0 1 B C m B X c Q f C˛ ./ C X ˛;n;j Œm C Œm B ıjQ;r B r;n .f I / f ./ C N B C Y @˛1 ;:::;˛N 0W jQD1 ˛k Š A j˛jDjQ kD1

X 1

0 @

˛1 ;:::;˛N 0 j˛jDm

Z

A

N Y RN

!r f1; C˛ ; n C !r f2; C˛ ; n ! N Y ˛k Š

1

kD1

! jsk j

˛k

kD1

Proof. By Theorem 10 of [1].

ksk2 1C n

r

! dn .s/ :

(5.22) t u

62


Also, we have Theorem5.7. Let f W RN ! C such that f D f1 C if2 : Here j D 1; 2. Let fj 2 CBl RN , l; N 2 N: The assumptions of Theorem 5.5 are valid for fj . Call D 0; ˇ: Assume Z ksk2 r dn .s/ < 1: 1C n RN Then

Z Œ0 f f r;n

ksk2 r 1C dn .s/ n RN 1 !r f1; ; n C !r f2; ; n :

(5.23)

t u

Proof. By Theorem 11 of [1]. By Theorem 4 of [4], we get

N Theorem N 5.8. Let f W R ! C W f D f1 C if2 : Here j D 1; 2. Let fj 2 m C R , m 2 N, N 1; with fj;˛ 2 Lp RN ; j˛j D m, x 2 RN . Let p; q > 1 W 1 C q1 D 1: Here n is a Borel probability measure on RN for n > 0, .n /n2N is p C a bounded PN sequence. Assume for all ˛ WD .˛1 ; : : : ; ˛N /, ˛k 2 Z , k D 1; : : : ; N; j˛j WD kD1 ˛k D m, we have that

Z RN

N Y

! jsk j

kD1

˛k

ksk2 1C n

r !p dn .s/ < 1:

(5.24)

C Q For PN j D 1; : : : ; m, and ˛ WD .˛1 ; : : : ; ˛N /, ˛k 2 Z , k D 1; : : : ; N; j˛j WD Q kD1 ˛k D j , call Z Y N c˛;n;jQ WD sk˛k dn .s/ : (5.25) RN kD1

Then

0 1 B C m B C X Œm B X c˛;n;jQ f˛ .x/ C Œm ıjQ;r B r;n .f I x/ f .x/ C N B C Y @j˛jDjQ jQD1 ˛k Š A kD1

0

!B BX B B 1 .q .m 1/ C 1/ q B @j˛jDm m

1

C C 1 C C ! N C Y ˛k Š A kD1

p;x

5.2 Main Results

63

#p r #p ksk 2 dn .s/ 1C jsk j˛k n RN kD1 !r .f1;˛ ; n /p C !r .f2;˛ ; n /p :

"Z

"

!

N Y

1

(5.26)

We further get N Theorem f1 C if2 ; j D 1; 2. Let fj 2 W R ! C W f D N 5.9. LetN f , N 1; p; q > 1 W p1 C q1 D 1: Assume n probability C R \ Lp R Borel measures on RN , .n /n2N > 0 and bounded. Also suppose

Z RN

ksk2 rp dn .s/ < 1: 1C n

(5.27)

Then Z Œ0 r;n .f / f

p1 ksk2 rp dn .s/ 1C p n RN !r .f1 ; n /p C !r .f2 ; n /p :

(5.28)

t u

Proof. By Theorem 6 of [4]. Based on Theorem 8 of [4], we get

N Theorem f W R ! C; f D f1 C if2 ; j D 1; 2. Let fj N2 N 5.10. Let N C R \ L1 R , N 1: Assume n probability Borel measures on R , .n /n2N > 0 and bounded. Also suppose

Z RN

ksk2 r dn .s/ < 1: 1C n

(5.29)

Then Z Œ0 r;n .f / f 1

RN

ksk2 r dn .s/ 1C n

(5.30)

.!r .f1 ; n /1 C !r .f2 ; n /1 / : Based on Theorem 10 of [4], we get

Theorem 5.11. Let f W RN ! C; f D f1 C if2 ; j D 1; 2. Let fj 2 C m RN , m; N 2 N; with fj;˛ 2 L1 RN ; j˛j D m, x 2 RN . Here, n is a Borel probability measure on RN for n > 0, .n /n2N is a bounded sequence. Assume for all PN ˛ WD .˛1 ; : : : ; ˛N /, ˛k 2 ZC , k D 1; : : : ; N; j˛j WD ˛ kD1 k D m that we have

64


Z

N Y

RN

!

ksk2 1C n

jsk j˛k

kD1

r ! dn .s/ < 1:

(5.31)


Then

0 1 B C m B C X Œm B X c˛;n;jQ f˛ .x/ C Œm ıjQ;r B r;n .f I x/ f .x/ C N B C Y @j˛jDjQ jQD1 ˛k Š A kD1

1;x

1

0 B XB B 1 B N BY j˛jDm @ ˛

k

C C C C !r .f1;˛ ; n /1 C !r .f2;˛ ; n /1 C ŠA

(5.33)

kD1

Z

N Y RN

kD1

! jsk j

˛k

ksk2 1C n

r dn .s/ :

(5.2)

Basedon Theorem 12 of [1], we get

Theorem 5.12. Let f W RN ! C; f D f1 C if2 I j D 1; 2, with fj 2 C mCl RN ; m; l; N 2 N: The of Theorem 5.5 are valid for fj . Call D 0; ˇ. Let assumptions fj;. C˛/ 2 Lp RN , j˛j D m, x 2 RN , p; q > 1 W p1 C q1 D 1: Here, n is a Borel probability measure on RN for n > 0, .n /n2N is a bounded P sequence. Assume for all ˛ WD .˛1 ; : : : ; ˛N /, ˛k 2 ZC , k D 1; : : : ; N; j˛j WD N kD1 ˛k D m we have that ! ! p Z N Y ksk2 r dn .s/ < 1: (5.34) 1C jsk j˛k n RN kD1

C Q For PN j D 1; : : : ; m, and ˛ WD .˛1 ; : : : ; ˛N /, ˛k 2 Z , k D 1; : : : ; N; j˛j WD Q kD1 ˛k D j , call Z Y N sk˛k dn .s/ : c˛;n;jQ WD RN kD1

5.2 Main Results

Then

65

1 0 C B m B X c Q f C˛ .x/ C X ˛;n;j C Œm Œm B ıjQ;r B C r;n .f I x/ f .x/ N C B Y A @j˛jDjQ jQD1 ˛k Š kD1

0

1

!B BX B B 1 B q .q .m 1/ C 1/ @j˛jDm m

(5.35) p;x

C C C ! C N C Y ˛k Š A 1

kD1

"

"Z RN

N Y

! jsk j

ksk2 1C n

˛k

kD1

# p1

r # p

dn .s/

!r f1; C˛ ; n p C !r f2; C˛ ; n p :

Based on Theorem 13 of [1] we get

Theorem 5.13. Let f W RN ! C; f D f1 C if2 ; j D 1; 2: Let fj 2 C l RN ; l; N 2 N: The assumptions of Theorem 5.5 are valid. Call D 0; ˇ. Let fj; 2 Lp RN , x 2 RN , p; q > 1 W p1 C q1 D 1: Assume n probability Borel measures on RN ; .n /n2N > 0 and bounded. Also suppose Z ksk2 rp dn .s/ < 1: 1C n RN Then Z Œ0 f f r;n

p

RN

p1 ksk2 rp dn .s/ 1C n

(5.36)

!r f1; ; n p C !r f2; ; n p : By Theorem 14 of [1], we get

Theorem 5.14. Let f W RN ! C; f D f1 C if2 ; j D 1; 2: Let fj 2 C l RN ; l; N 2 N: The assumptions of Theorem 5.5 are valid for fj . Call D 0; ˇ. Let fj; 2 L1 RN , x 2 RN : Assume n probability Borel measures on RN ; .n /n2N > 0 and bounded. Also suppose Z ksk2 r 1C dn .s/ < 1: n RN

66


Then Z Œ0 f f r;n

ksk2 r dn .s/ 1C n RN 1 !r f1; ; n 1 C !r f2; ; n 1 :

(5.37)

Finally, we have

Theorem 5.15. Let f W RN ! C; f D f1 C if2 ; j D 1; 2, with fj 2 C mCl RN , m; l; N 2 N. The of Theorem 5.5 are valid for fj . Call D 0; ˇ. Let assumptions fj;. C˛/ 2 L1 RN ; j˛j D m, x 2 RN . Here n is a Borel probability measure on C RN for n > 0, .n /n2N PNis bounded. Assume for all ˛ WD .˛1 ; : : : ; ˛N /, ˛k 2 Z , k D 1; : : : ; N; j˛j WD kD1 ˛k D m; we have that Z

N Y

RN

! jsk j

kD1

˛k

ksk2 1C n

r ! dn .s/ < 1:

(5.38)


Then 0 1 B C m B C X Œm B X c˛;n;jQf. C˛/ .x/ C Œm ıjQ;r B C r;n .f I x/ f .x/ N B C Y @j˛jDjQ A jQD1 ˛k Š kD1

B XB B 1 B N BY j˛jDm @ ˛

k

C C C C !r f1; C˛ ; n 1 C !r f2; C˛ ; n 1 C ŠA

kD1 N Y RN

1;x

1

0

Z

(5.40)

! ˛k

jsk j

kD1

Proof. By Theorem 15 of [1].

ksk2 1C n

r dn .s/ : t u

5.3 Applications

67

5.3 Applications Let all entities as in Sect. 5.2. We define the following specific operators for f W RN ! C: (a) The general multivariate Picard singular integral operators: Œm Pr;n

.f I x1 ; : : : ; xN / WD

r X

1 N

.2n /

Œm

˛j;r

0

1

B B @

jsi jC A

N X

Z RN

(5.41)

j D0 C

i D1 n


ds1 : : : dsN :

(b) The general multivariate Gauss–Weierstrass singular integral operators: r X 1 Œm Œm .f I x1 ; : : : ; xN / WD p ˛j;r Wr;n N j D0 n 0

1

B B @

C si2 C A

N X

Z RN

i D1 n


(5.42)

ds1 : : : dsN :

(c) The general multivariate Poisson–Cauchy singular integral operators: r X

Œm .f I x1 ; : : : ; xN / WD WnN Ur;n

Œm

˛j;r

(5.43)

j D0

Z RN

f .x1 C s1 j; : : : ; xN C sN j /

with ˛ 2 N, ˇ >

1 ; 2˛

N Y

1

2˛ i D1 si

C n2˛

ˇ ds1 : : : dsN ;

and 2˛ˇ1

Wn WD

.ˇ/ ˛n 1 2˛ ˇ

1 2˛

:

(5.44)

(d) The general multivariate trigonometric singular integral operators: Œm .f I x1 ; : : : ; xN / WD N Tr;n n

r X j D0

Œm

˛j;r

(5.45)

68


Z RN

f .x1 C s1 j; : : : ; xN C sN j /

N Y

0 @

i D1

sin

12ˇ si n

si

A

ds1 : : : dsN ;

where ˇ 2 N, and

n WD

2n12ˇ

ˇ

.1/ ˇ

ˇ X kD1

.1/k

k 2ˇ1 : .ˇ k/Š .ˇ C k/Š

(5.46)

Œm Œm Œm Œm One can apply the results of this chapter to the operators Pr;n ; Wr;n ; Ur;n , Tr;n Œm (special cases of r;n ) and derive interesting results.

5.4 Conclusion Our approximation results here imply important convergence properties of operators Œm r;n to the unit operator.

References 1. G. Anastassiou, Global smoothness preservation and simultaneous approximation for multivariate general singular integral operators, Applied Math. Letters, 24(2011), 1009-1016. 2. G. Anastassiou, Multivariate Complex General Singular Integral Operators Simultaneous Approximation, Journal of Concrete and Applicable Mathematics, accepted, 2010. 3. G. Anastassiou, General uniform Approximation theory by multivariate singular integral operators, submitted, 2011. 4. G. Anastassiou, Lp -general approximations by multivariate singular integral operators, submitted, 2011.

Chapter 6

Approximation of Functions of Two Variables via Almost Convergence of Double Sequences

The idea of almost convergence for double sequences was introduced by Moricz and Rhoades [6]. In this chapter, we use this concept to prove a Korovkin-type approximation theorem for functions of two variables and we give an example. Furthermore, we present the consequences of the main theorem. This chapter is based on [1].

6.1 Introduction and Preliminaries Let c and l1 denote the spaces of all convergent and bounded sequences, respectively, and note that c l1 . In the theory of sequence spaces, a beautiful application of the well-known Hahn–Banach Extension Theorem gave rise to the concept of the Banach limit. That is, the lim functional defined on c can be extended to the whole of l1 and this extended functional is known as the Banach limit [2], which was used by Lorentz [5] to define a new type of convergence, known as the almost convergence. A double sequence x D .xjk / of real or complex numbers is said to be bounded if kxk1 D supj;k jxjk j < 1: The space of all bounded double sequences is denoted by Mu . A double sequence x D .xjk / is said to converge to the limit L in Pringsheim’s sense (shortly, p-convergent to L) [10] if for every " > 0 there exists an integer N such that jxjk Lj < " whenever j; k > N . In this case L is called the p-limit of x. If in addition x 2 Mu , then x is said to be boundedly convergent to L in Pringsheim’s sense (shortly, bp-convergent to L).


69

70

6

Approximation of Functions of Two Variables

Let denote the vector space of all double sequences with the vector space operations defined coordinatewise. Vector subspaces of are called double sequence spaces. In addition to the above-mentioned double sequence spaces, we consider the double sequence space Lu WD

8 < :

x 2 j kxk1 WD

X

jxjk j < 1

j;k

9 = ;

of all absolutely summable double sequences. All considered double sequence spaces are supposed to contain ˆ WD span fejk j j; k 2g; where

1 I if .j; k/ D .i; `/; 0 I otherwise: P P P We denote the pointwise sums j;k ejk , j ejk .k 2/, and k ejk .j 2/ by e, ek and ej respectively. The idea of almost convergence for double sequences was introduced and studied by Moricz and Rhoades [6]. A double sequence x D .xjk / of real numbers is said to be almost convergent to a limit L if ˇ ˇ ˇ ˇ X nCq1 X ˇ ˇ 1 mCp1 lim sup ˇˇ () xjk Lˇˇ D 0 p;q!1 m;n>0 ˇ pq ˇ j Dm kDn jk eil

D

In this case, L is called the F2 -limit of x and we shall denote by F2 the space of all almost convergent double sequences. Note that a convergent double sequence need not be almost convergent. However, every bounded convergent double sequence is almost convergent and every almost convergent double sequence is bounded. Example 6.1. The double sequence z D .zmn / defined by

zmn

8 1 if m D n odd; ˆ ˆ < D 1 if m D n even; ˆ ˆ : 0 .m 6D n/I

(6.1)

is almost convergent to zero but not convergent. For recent developments on almost convergent double sequences and matrix transformations, we refer to [7–9, 11]. If m D n D 1 in ./, then we get .C; 1; 1/-convergence, and in this case we write xjk ! `.C; 1; 1/I where ` D .C; 1; 1/-lim x.

6.2 Korovkin-Type Approximation Theorem

71

Let C Œa; b be the space of all functions f continuous on Œa; b. We know that C Œa; b is a Banach space with norm kf k1 WD sup jf .x/j; f 2 C Œa; b: x2Œa;b

The classical Korovkin approximation theorem states as follows [4]: Let .Tn / be a sequence of positive linear operators from C Œa; b into C Œa; b: Then limn kTn .f; x/ f .x/k1 D 0, for all f 2 C Œa; b if and only if limn kTn .fi ; x/ fi .x/k1 D 0, for i D 0; 1; 2, where f0 .x/ D 1; f1 .x/ D x and f2 .x/ D x 2 : Quite recently, such type of approximation theorems are proved in [3] for functions of two variables by using statistical convergence. In this chapter, we use the notion of almost convergence to prove approximation theorems for functions of two variables.

6.2 Korovkin-Type Approximation Theorem The following is the F2 -version of the classical Korovkin approximation theorem followed by an example to show its importance. Let C.I 2 / be the space of all two dimensional continuous functions on I I , where I D Œa; b. Suppose that Tm;n W C.I 2 / ! C.I 2 /. We write Tm;n .f I x; y/ for Tm;n .f .s; t/I x; y/I and we say that T is a positive operator if T .f I x; y/ 0 for all f .x; y/ 0. Theorem 6.2. Let .Tj;k / be a double sequence of positive linear operators from PmCp1 PnCq1 1 C.I 2 / into C.I 2 / and Dm;n;p;q .f I x; y/ D pq Tj;k .f I x; y/: j Dm kDn Then for all f 2 C.I 2 / F2 - lim Tj;k .f I x; y/ f .x; y/ j;k!1

.f I x; y/ f .x; y/ D lim m;n;p;q p;q!1

D 0; i:e: 1

D 0; uniformly in m; n:

(6.2)

1

if and only if lim Dm;n;p;q .1I x; y/ 1

D 0 uniformly in m; n;

(6.3)

lim Dm;n;p;q .sI x; y/ x

D 0 uniformly in m; n;

(6.4)

p;q!1

1

p;q!1

1

72

6


lim Dm;n;p;q .tI x; y/ y p;q!1

2 2 2 2 lim 4Dm;n;p;q .s C t I x; y/ .x C y /

p;q!1

D 0 uniformly in m; n:

(6.5)

D 0 uniformly in m; n:

(6.6)

1

1

Proof. Since each 1; x; y; x 2 C y 2 belongs to C.I 2 /; conditions (6.3)–(6.6) follow immediately from (6.2). By the continuity of f on I 2 , we can write jf .x; y/j M; 1 < x; y < 1; where M D kf k1 : Therefore, jf .s; t/ f .x; y/j 2M; 1 < s; t; x; y < 1:

(6.7)

Also, since f 2 C.I 2 /; for every > 0, there is ı > 0 such that jf .s; t/ f .x; y/j < ; 8 js xj < ı and jt yj < ı: Using (6.7), (6.8), putting we get

1

D

jf .s; t/ f .x; y/j < C

1 .s; x/

2M . ı2

1

D .s x/2 and

C

2 /;

2

D

(6.8)

2 .t; y/

D .t y/2 ,

8 js xj < ı and jt yj < ı:

This is,

2M . ı2

1

C

2/

< f .s; t/ f .x; y/ < C

2M . ı2

1

C

2 /:

Now, operating Tj;k .1I x; y/ to this inequality since Tj;k .f I x; y/ is monotone and linear. We obtain 2M Tj;k .1I x; y/ 2 . 1 C 2 / < Tj;k .1I x; y/.f .s; t/ f .x; y// ı 2M < Tj;k .1I x; y/ C 2 . 1 C 2 / : ı Note that x and y are fixed and so f .x; y/ is constant number. Therefore Tj;k .1I x; y/

2M Tj;k . ı2

1

C

2 I x; y/

< Tj;k .f I x; y/ f .x; y/Tj;k .1I x; y/ < Tj;k .1Ix;y/C

2M Tj;k . ı2

1 C 2 Ix;y/:

(6.9)


73

But Tj;k .f I x; y/ f .x; y/ D Tj;k .f I x; y/ f .x; y/Tj;k .1I x; y/ C f .x; y/Tj;k .1I x; y/ f .x; y/ D ŒTj;k .f I x; y/ f .x; y/Tj;k .1I x; y/ C f .x; y/ŒTj;k .1I x; y/ 1:

(6.10)

Using (6.9) and (6.10), we have Tj;k .f I x; y/ f .x; y/ < Tj;k .1I x; y/ C

2M Tj;k . ı2

1

C

2 I x; y/

C f .x; y/.Tj;k .1I x; y/ 1/:

(6.11)

Now Tj;k .

1

C

2 I x; y/

D Tj;k ..s x/2 C .t y/2 I x; y/ D Tj;k .s 2 2sx C x 2 C t 2 2ty C y 2 I x; y/ D Tj;k .s 2 C t 2 I x; y/ 2xTj;k .sI x; y/ 2yTj;k .tI x; y/ C .x 2 C y 2 /Tj;k .1I x; y/ D ŒTj;k .s 2 C t 2 I x; y/ .x 2 C y 2 / 2xŒTj;k .sI x; y/ x 2yŒTj;k .tI x; y/ y C .x 2 C y 2 /ŒTj;k .1I x; y/ 1:

Using (6.11), we obtain Tj;k .f I x; y/ f .x; y/ < Tj;k .1I x; y/ C

2M fŒTj;k .s 2 C t 2 I x; y/ .x 2 C y 2 / ı2

2xŒTj;k .sI x; y/ x 2yŒTj;k .tI x; y/ y C .x 2 C y 2 /ŒTj;k .1I x; y/ 1g C f .x; y/.Tj;k .1I x; y/ 1/ D ŒTj;k .1I x; y/ 1 C C

2M fŒTj;k .s 2 C t 2 I x; y/ ı2

.x 2 C y 2 / 2xŒTj;k .sI x; y/ x 2yŒTj;k .tI x; y/ y C .x 2 C y 2 /ŒTj;k .1I x; y/ 1g C f .x; y/.Tj;k .1I x; y/ 1/:

74

6


Since is arbitrary, we can write Tj;k .f I x; y/ f .x; y/ ŒTj;k .1I x; y/ 1 C

2M fŒTj;k .s 2 C t 2 I x; y/ ı2

.x 2 C y 2 / 2xŒTj;k .sI x; y/ x 2yŒTj;k .tI x; y/ y C .x 2 C y 2 /ŒTj;k .1I x; y/ 1g C f .x; y/.Tj;k .1I x; y/ 1/: Similarly, Dm;n;p;q .f I x; y/ f .x; y/ ŒDm;n;p;q .1I x; y/ 1 C

2M ˚ ŒDm;n;p;q .s 2 C t 2 I x; y/ .x 2 C y 2 / ı2 2xŒDm;n;p;q .sI x; y/ x 2yŒDm;n;p;q .tI x; y/ y C.x 2 C y 2 /ŒDm;n;p;q .1I x; y/ 1

Cf .x; y/.Dm;n;p;q .1I x; y/ 1/ and therefore 2 2 Dm;n;p;q .f I x; y/ f .x; y/ C 2M.a C b / C M ı2 1 Dm;n;p;q .1I x; y/ 1

1

4M a 2 Dm;n;p;q .sI x; t/ x ı 1 4M b 2 Dm;n;p;q .tI x; y/ y ı 1 2M 2 2 2 2 C 2 Dm;n;p;q .s C t I x; y/ .x C y / : ı 1


75

Letting p; q ! 1 and using (6.3), (6.4), (6.5), and (6.6), we get D 0; uniformly in m; n: .f I x; y/ f .x; y/ D lim m;n;p;q p;q!1 1

t u

This completes the proof of the theorem.

In the following, we give an example of a double sequence of positive linear operators satisfying the conditions of Theorem 6.2 but does not satisfy the conditions of the Korovkin theorem. Example 6.3. Consider the sequence of classical Bernstein polynomials of two variables ! ! m X n X j k m n j f Bm;n .f I x; y/ WD x .1 x/mj ; j k m n j D0 kD0

y k .1 y/nk I 0 x; y 1: Let Pm;n W C.I 2 / ! C.I 2 / be defined by Pm;n .f I x; y/ D .1 C zmn /Bm;n .f I x; y/; where .zmn / is a double sequence defined as above. Then Bm;n .1I x; y/ D 1; Bm;n .sI x; y/ D x; Bm;n .tI x; y/ D y; Bm;n .s 2 C t 2 I x; y/ D x 2 C y 2 C

y y2 x x2 C ; m n

and a double sequence .Pm;n / satisfies the conditions (6.3), (6.4), (6.5) and (6.6). Hence, we have F2 - lim kPm;n .f I x; y/ f .x; y/k1 D 0: m;n!1

On the other hand, we get Pm;n .f I 0; 0/ D .1 C zmn /f .0; 0/, since Bm;n .f I 0; 0/ D f .0; 0/, and hence kPm;n .f I x; y/ f .x; y/k1 jPm;n .f I 0; 0/ f .0; 0/j D zmn jf .0; 0/j: We see that .Pm;n / does not satisfy the classical Korovkin theorem, since lim zmn does not exist.

m;n!1

76

6


6.3 Some Consequences Now we present here some consequences of Theorem 6.2. Theorem 6.4. Let .Tm;n / be a double sequence of positive linear operators on C.I 2 / such that lim kTmC1;nC1 Tm;nC1 TmC1;n C Tm;n k D 0:

(6.12)

F2 - lim kTm;n .t I x; y/ t k1 D 0 . D 0; 1; 2; 3/;

(6.13)

m;n

If m;n

where t0 .x; y/ D 1, t1 .x; y/ D x, t2 .x; y/ D y and t3 .x; y/ D x 2 C y 2 . Then for any function f 2 C.I 2 /, we have lim kTm;n .f I x; y/ f .x; y/k1 D 0: m;n

(6.14)

Proof. From Theorem 6.2, we have that if (6.13) holds then lim kDm;n;p;q .f I x; y/ f .x; y/k1 D 0; uniformly in m; n: p;q

(6.15)

We have the following inequality kTm;n .f I x; y/ f .x; y/k1 kDm;n;p;q .f I x; y/ f .x; y/k1 mCp1 nCq1 1 X X C pq

j X

k X

! kT˛;ˇ T˛1;ˇ T˛;ˇ1 C T˛1;ˇ1 k

j DmC1 kDnC1 ˛DmC1 ˇDnC1

kDm;n;p;q .f I x; y/ f .x; y/k1 p1q1 C sup kTj;k Tj 1;k Tj;k1 C Tj 1;k1 k : 2 2 j m;kn Hence using (6.12) and (6.15), we get (6.14). This completes the proof of the theorem.

(6.16)

t u

We know that double almost convergence implies .C; 1; 1/ convergence. This motivates us to further generalize our main result by weakening the hypothesis or to add some condition to get more general result. Theorem 6.5. Let .Tm;n / be a double sequence of positive linear operators on C.I 2 / such that .C; 1; 1/ lim kTm;n .t ; x/ t k1 D 0 . D 0; 1; 2; 3/; m;n

(6.17)

6.3 Some Consequences

77

and lim p;q

sup mp;nq

mn mCp1;nCq1 .f I x; y/ m1;n1 .f I x; y/ D 0; (6.18) pq 1

where

XX 1 Tj;k .f I x; y/: .m C 1/.n C 1/ m

m;n .f I x; y/ D

n

j D0 kD0

Then for any function f 2 C.I /, we have 2

F2 - lim Tm;n .f I x; y/ f .x; y/ D 0: m;n!1 1

Proof. For m p 1I n q 1, it is easy to show that Dm;n;p;q .f I x; y/ D mCp1;nCq1 .f I x; y/ C

mn .mCp1;nCq1 .f I x; y/ pq

m1;n1 .f I x; y//; which implies sup Dm;n;p;q .f I x; y/ mCp1;nCq1 .f I x; y/

mp;nq

D

sup mp;nq

1

mn mCp1;nCq1 .f I x; y/ m1;n1 .f I x; y// : pq 1

(6.19)

Also by Theorem 6.2, Condition (6.17) implies that .C; 1; 1/- lim Tm;n .f I x; y/ f .x; y/ D 0: m;n!1

(6.20)

1

Using (6.17)–(6.20) and the fact that almost convergence implies .C; 1; 1/ convergence, we get the desired result. This completes the proof of the theorem. t u Theorem 6.6. Let .Tm;n / be a double sequence of positive linear operators on C.I 2 / such that lim sup m;n s;t

sCm1 t Cn1 1 X X kTm;n Tj;k k D 0: mn j Ds kDt

78

6


If F2 - lim kTm;n .t ; x/ t k1 D 0 m;n

. D 0; 1; 2; 3/:

(6.21)

Then for any function f 2 C.I 2 /, we have lim kTm;n .f I x; y/ f .x; y/k1 D 0: m;n

(6.22)

Proof. From Theorem 6.2, we have that if (6.21) holds then F2 - lim kTm;n .f I x; y/ f .x; y/k1 D 0; m;n

which is equivalent to lim sup kDs;t;m;n .f I x; y/ f .x; y/k1 D 0: m;n s;t

(6.23)

Now Tm;n Ds;t;m;n D Tm;n

sCm1 t Cn1 1 X X Tj;k mn j Ds kDt

D

sCm1 t Cn1 1 X X .Tm;n Tj;k /: mn j Ds kDt

Therefore, sup kTm;n Ds;t;m;n k1 sup s;t

s;t

sCm1 t Cn1 1 X X kTm;n Tj;k k: mn j Ds kDt

Now, by using the hypothesis we get lim sup kTm;n .f I x; y/ Ds;t;m;n .f I x; y/k1 D 0: m;n s;t

(6.24)

By the triangle inequality, we have kTm;n .f I x; y/ f .x; y/k1 kTm;n .f I x; y/ Ds;t;m;n .f I x; y/k1 CkDs;t;m;n .f I x; y/ f .x; y/k1 ; and hence from (6.23) and (6.24), we get lim kTm;n .f I x; y/ f .x; y/k1 D 0; m;n

that is (6.22) holds. This completes the proof of the theorem.

t u

References

79

References 1. G. Anastassiou, M. Mursaleen and S. Mohiuddine, Some Approximation Theorems for Functions of Two Variables through Almost Convergence of Double Sequences, Journal of Computational Analysis and Applications, Vol. 13, No. 1(2011), 37-46. 2. S. Banach, Théorie des Operations Lineaires, Warszawa, 1932. 3. F. Dirik and K. Demirci, Korovkin type approximation theorem for functions of two variables in statistical sense, Turk. J. Math. 33(2009)1-11. 4. A.D. Gad˘ziev, The convergence problems for a sequence of positive linear operators on unbounded sets, and theorems analogous to that of P.P. Korovkin, Soviet Math. Dokl. 15(1974)1433-1436. 5. G.G. Lorentz, A contribution to theory of divergent sequences, Acta Math. 80(1948)167-190. 6. F. Moricz and B.E. Rhoades, Almost convergence of double sequences and strong regularity of summability matrices, Math. Proc. Camb. Phil. Soc. 104(1988)283-294. 7. M. Mursaleen, Almost strongly regular matrices and a core theorem for double sequences, J. Math. Anal. Appl. 293(2004)523-531. 8. M. Mursaleen and O.H.H. Edely, Almost convergence and a core theorem for double sequences, J. Math. Anal. Appl. 293(2004)532-540. 9. M. Mursaleen and S.A. Mohiuddine, Almost bounded variation of double sequences and some four dimensional summability matrices, Publ. Math. Debrecen 75(2009)495-508. 10. A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Z. 53(1900) 289-321. 11. M. Zeltser, M. Mursaleen and S.A. Mohiuddine, On almost conservative matrix methods for double sequence spaces, Publ. Math. Debrecen 75(2009)387-399.

Approximation by Multivariate Singular Integrals

Approximation by Multivariate Singular Integrals

Singular Integrals

Multivariate approximation theory

Multivariate Approximation and Applications

Multivariate approximation theory

Singular Integrals and Related Topics

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Multivariate approximation theory: selected topics

Singular integrals and related topics

An Introduction to Singular Integrals

Multidimensional singular integrals and integral equations

Singular Integrals and Differentiability Properties of Functions

Clifford Wavelets, Singular Integrals, and Hardy Spaces

Multidimensional Singular Integrals and Integral Equations

Approximation by Spline Functions

Approximation by algebraic numbers

Approximation by spline functions

Singular integrals and differentiability properties of functions

Singular integrals and rectifiable sets in Rn

A Besov Estimate for Multilinear Singular Integrals

Approximation by algebraic numbers

Topics in multivariate approximation and interpolation

Interpolation and Approximation by Polynomials

Interpolation and Approximation by Polynomials

Normal approximation by Stein's method

Interpolation and Approximation by Polynomials

Wavelets and Singular Integrals on Curves and Surfaces

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Asymptotic Approximation of Integrals (Classics in Applied Mathematics)

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