Approximation Theory: Moduli of Continuity and Global Smoothness Preservation

519 Pages·2000·14.25 MB·English

by  George A. Anastassiou

To the memory of our parents:
Angelos and Panagiota Anastassiou
and
Gheorghe and Ana Gal
George A. Anastassiou
Sorin G. Gal
Approximation Theory
Moduli of Continuity and Global
Smoothness Preservation
Springer Science+Business Media, LLC
George A. Anastassiou Sorin G. Gal
Department of Mathematical Sciences Department of Mathematics
University of Memphis University of Oradea
Memphis, TN 38152 3700 Oradea
USA Romania
Ubrary of Congress Cataloging-in-Publication Data
Anastassiou, George A., 1952-
Approximation theory : moduli of continuity and global smoothness preservation/
George A. Anastassiou, Sorin G. Gal.
p.cm.
Includes bibliographical references and index.
ISBN 978-1-4612-7112-3 ISBN 978-1-4612-1360-4 (eBook)
DOI 10.1007/978-1-4612-1360-4
1. Smoothness of functions. 2. Moduli theory. 3. Approximation theory. 1. Gal, Sorin
G., 1953-11. Title.
QA355.A53 2000
511'.4-dc21 99-057004
CIP
AMS Subject Classifications: 41-XX, 42AI0, 49J22, 62L20
Printed on acid-free paper.
©2000 Springer Science+Business Media New York
OriginaIly published by Birkhiiuser Boston in 2000
Softcover reprint of the hardcover 1s I edition 2000
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ISBN 978-1-4612-7112-3 SPIN 10740157
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Cover design by Jeff Cosloy, Newton, MA.
9 8 7 6 5 432 1
Preface
We study in Part I of this monograph the computational aspect of almost
all moduli of continuity over wide classes of functions exploiting some of
their convexity properties. To our knowledge it is the first time the entire
calculus of moduli of smoothness has been included in a book. We then
present numerous applications of Approximation Theory, giving exact val
ues of errors in explicit forms. The K-functional method is systematically
avoided since it produces nonexplicit constants. All other related books so
far have allocated very little space to the computational aspect of moduli
of smoothness.
In Part II, we study/examine the Global Smoothness Preservation Prop
erty (GSPP) for almost all known linear approximation operators of ap
proximation theory including: trigonometric operators and algebraic in
terpolation operators of Lagrange, Hermite-Fejer and Shepard type, also
operators of stochastic type, convolution type, wavelet type integral opera
tors and singular integral operators, etc. We present also a sufficient general
theory for GSPP to hold true. We provide a great variety of applications
of GSPP to Approximation Theory and many other fields of mathemat
ics such as Functional analysis, and outside of mathematics, fields such as
computer-aided geometric design (CAGD). Most of the time GSPP meth
ods are optimal. Various moduli of smoothness are intensively involved in
Part II. Therefore, methods from Part I can be used to calculate exactly the
error of global smoothness preservation. It is the first time in the literature
that a book has studied GSPP.
This monograph contains the research of both authors over the past ten
years in these subjects. It also references most of the works of other main
vi Preface
researchers in these areas. The proving methods are rather elementary, that
is, making this material accessible to graduate students and researchers.
This book is intended for use in the fields of approximation theory, pure
mathematics, applied mathematics, probability, numerical analysis, and en
gineering researchers. It is also suitable for graduate courses in the above
disciplines.
We would like to thank Diane M. Mittelmeier of the University of Mem
phis for being our internet-Latex operator. Also, we wish to thank our typ
ists for doing a great job so punctually: Kate MacDougall of Warren, RI
and Georgeta Bonda of Babes-Bolyai University, Cluj-Napoca, Romania.
At last, but not least, we would like to thank Professor Michiel Hazewinkel
of CWI, Amsterdam, the Netherlands, for giving us great encouragement
throughout this major project.
All the communications of the two authors in writing this monograph
were entirely based on e-mail.
February 1, 1999
George A. Anastassiou Sorin G. Gal
Memphis, U.S.A. Oradea, Romania
Contents
Preface v
1 Introduction 1
1.1 On Chapter 2: Uniform Moduli of Smoothness 1
1.2 On Chapter 3: LP-Moduli of Smoothness, 1 ::; p < 00 . 8
1.3 On Chapter 4: Moduli of Smoothness of Special Type 11
1.4 On Chapter 5: Global Smoothness Preservation
by Trigonometric Operators . . . . . . . . . . . 14
1.5 On Chapter 6: Global Smoothness Preservation
by Algebraic Interpolation Operators . . . . . . 16
1.6 On Chapter 7: Global Smoothness Preservation
by General Operators .............. 19
1. 7 On Chapter 8: Global Smoothness Preservation
by Multivariate Operators . . . . . . . . . . . . . . . . . .. 20
1.8 On Chapter 9: Stochastic Global Smoothness Preservation. 22
1.9 On Chapter 10: Shift Invariant Univariate
Integral Operators . . . . . . . . . . . . . . . . . . . . . .. 24
1.10 On Chapter 11: Shift Invariant Multivariate
Integral Operators . . . . . . . . . . . . . . . . . . . . .. 26
1.11 On Chapter 12: Differentiated Shift Invariant Univariate
Integral Operators . . . . . . . . . . . . . . . . . . . . .. 28
1.12 On Chapter 13: Differentiated Shift Invariant Multivariate
Integral Operators . . . . . . . . . . . . . . . . . . . . . .. 31
viii Contents
1.13 On Chapter 14: Generalized Shift Invariant Univariate
Integral Operators . . . . . . . . . . . . . . . . . . . . 33
1.14 On Chapter 15: Generalized Shift Invariant Multivariate
Integral Operators . . . . . . . . . . . . . . . . . . . . 35
1.15 On Chapter 16: General Theory of Global Smoothness
Preservation by Univariate Singular Integrals . . . . . 38
1.16 On Chapter 17: General Theory of Global Smoothness
Preservation by Multivariate Singular Integrals . . . . 41
1.17 On Chapter 18: Gonska Progress in Global Smoothness
Preservation. . . . . . . . . . . . . . . . . . . . . . 45
1.18 On Chapter 19: Miscellaneous Progress on Global
Smoothness Preservation. . . . . . . . . . . . . . . . . . .. 45
1.19 On Chapter 20: Other Applications of the Global
Smoothness Preservation Property 45
1.20 Some History of GSPP . 46
1.21 Conclusion ............. 50
Part I Calculus of the Moduli of Smoothness
in Classes of Functions 55
2 Uniform Moduli of Smoothness 57
2.1 Modulus of Smoothness for Nonperiodic Functions
of One Variable . . . . . . . . . . . . . . . . . . 57
2.2 Modulus of Smoothness for Periodic Functions 75
2.3 Bivariate Modulus of Smoothness . . . 80
2.4 Ditzian-Totik Modulus of Smoothness .... 89
2.5 Applications. . . . . . . . . . . . . . . . . . . 104
2.6 Bibliographical Remarks and Open Problems 143
3 LP -Moduli of Smoothness, 1 :::; P < +00 145
3.1 Usual LP-Modulus of Smoothness .... 145
3.2 Averaged LP-Modulus of Smoothness .. 154
3.3 Ditzian-Totik LP-Modulus of Smoothness 163
3.4 Applications. . . . . . . . . . . . . . . . . 165
3.5 Bibliographical Remarks and Open Problems 169
4 Moduli of Smoothness of Special Type 171
...
4.1 One-Sided Modulus of Smoothness 171
4.2 Hausdorfl'-Sendov Modulus of Continuity. 176
4.3 An Algebraic Modulus of Smoothness 184
.....
4.4 Weighted Moduli of Smoothness 187
4.5 Applications. . . . . . . . . . . . . . . . . 193
4.6 Bibliographical Remarks and Open Problems 196
Contents ix
Part II Global Smoothness Preservation
by Linear Operators 201
5 Global Smoothness Preservation by
Trigonometric Operators 203
5.1 General Results. . . . . . . . . . . . . . . . . . . . . 203
5.2 Global Smoothness Preservation by Some Concrete
Trigonometric Operators . . . . . . . . . . . . . . . 205
5.3 Global Smoothness Preservation by Trigonometric
Projection Operators. . . . . . . . . . . . . . 208
5.4 Bibliographical Remarks and Open Problems 210
6 Global Smoothness Preservation by
Algebraic Interpolation Operators 211
6.1 Negative Results . . . . . . . . . . . . . . . . . . . . 211
6.2 Global Smoothness Preservation by Some Lagrange,
Hermite-Fejer and Shepard Operators . . . . . . . . 214
6.3 Global Smoothness Preservation by
Algebraic Projection Operators . . . . . . . . . . . . . . . . 224
6.4 Global Smoothness Preservation by Algebraic Polynomials
of Best Approximation . . . . . . . . . . . . . 227
6.5 Bibliographical Remarks and Open Problems . . . . . . . . 230
7 Global Smoothness Preservation by
General Operators 231
7.1 Introduction... 231
7.2 General Results. 233
7.3 Applications ... 241
7.3.1 Variation-Diminishing Splines. 241
7.3.2 Operators of Kratz and Stadtmiiller 243
7.4 Optimality of the Preceding Results 244
8 Global Smoothness Preservation by
Multivariate Operators 251
8.1 Introduction ................... . 251
8.2 A General Result for Operators Possessing the
Splitting Property . . . . . . . . . . 253
8.3 Bernstein Operators over Simplices . . . 254
8.4 Tensor Product Bernstein Operators . . 256
8.5 An Identity Between K-Functionals and
More Results on Global Smoothness 258
8.6 Example: A Comparison Theorem in
Stochastic Approximation . . . . . . 260
x Contents
9 Stochastic Global Smoothness Preservation 265
9.1 Introduction........................... 265
9.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 266
9.3 A Theorem on Stochastic Global Smoothness Preservation. 267
9.4 Applications.......................... 268
9.4.1 Stochastic Convolution-Type Operators on Cg[a, b] 268
9.4.2 Operators on Co [a, b] ....... 275
9.4.3 More Convolution-Type Operators . . . 277
10 Shift Invariant Univariate Integral Operators 279
10.1 Introduction ... 279
10.2 General Theory. 281
10.3 Applications. . . 287
11 Shift Invariant Multivariate Integral Operators 297
11.1 General Results . 297
11.2 Applications ......... . 312
12 Differentiated Shift Invariant
Univariate Integral Operators 325
12.1 Introduction ....... . 325
12.1.1 Other Motivations 326
12.2 General Results. 328
12.3 Applications. . . . . . . . 333
13 Differentiated Shift Invariant
Multivariate Integral Operators 347
13.1 Introduction ... 347
13.2 General Results . 350
13.3 Applications. . . 357
14 Generalized Shift Invariant
Univariate Integral Operators 373
14.1 General Theory. 373
14.2 Applications ....... . 382
15 Generalized Shift Invariant
Multivariate Integral Operators 391
15.1 General Theory. 391
15.2 Applications. . . . . . . . . . . 399
16 General Theory of Global Smoothness Preservation by
Univariate Singular Operators 401
16.1 Introduction. . . 401
16.2 General Theory . . . . . . . . 407