Approximation By Complex Bernstein And Convolution Type Operators

by Sorin G Gal

Overview

The monograph, as its first main goal, aims to study the overconvergence phenomenon of important classes of Bernstein-type operators of one or several complex variables, that is, to extend their quantitative convergence properties to larger sets in the complex plane rather than the real intervals. The operators studied are of the following types: Bernstein, Bernstein—Faber, Bernstein-Butzer, q-Bernstein, Bernstein-Stancu, Bernstein-Kantorovich, Favard-Szász-Mirakjan, Baskakov and Balázs-Szabados.The second main objective is to provide a study of the approximation and geometric properties of several types of complex convolutions: the de la Vallée Poussin, Fejér, Riesz-Zygmund, Jackson, Rogosinski, Picard, Poisson-Cauchy, Gauss-Weierstrass, q-Picard, q-Gauss-Weierstrass, Post-Widder, rotation-invariant, Sikkema and nonlinear. Several applications to partial differential equations (PDEs) are also presented.Many of the open problems encountered in the studies are proposed at the end of each chapter. For further research, the monograph suggests and advocates similar studies for other complex Bernstein-type operators, and for other linear and nonlinear convolutions.

Product Details

ISBN-13:	9789814282420
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	08/12/2009
Series:	Series On Concrete And Applicable Mathematics , #8
Pages:	352
Product dimensions:	6.60(w) x 9.80(h) x 1.10(d)

Table of Contents

Preface vii

1 Bernstein-Type Operators of One Complex Variable 1

1.0 Auxiliary Results in Complex Analysis 1

1.1 Bernstein Polynomials 6

1.1.1 Bernstein Polynomials on Compact Disks 6

1.1.2 Bernstein-Faber Polynomials on Compact Sets 19

1.2 Iterates of Bernstein Polynomials 26

1.3 Generalized Voronovskaja Theorems for Bernstein Polynomials 35

1.4 Butzer's Linear Combination of Bernstein Polynomials 42

1.5 q-Bernstein Polynomials 50

1.6 Bernstein-Stancu Polynomials 67

1.7 Bernstein-Kantorovich Type Polynomials 96

1.8 Favard-Szasz-Mirakjan Operators 103

1.9 Baskakov Operators 124

1.10 Balazs-Szabados Operators 139

1.11 Bibliographical Notes and Open Problems 149

2 Bernstein-Type Operators of Several Complex Variables 155

2.1 Introduction 155

2.2 Bernstein Polynomials 156

2.3 Favard-Szasz-Mirakjan Operators 166

2.4 Baskakov Operators 172

2.5 Bibliographical Notes and Open Problems 179

3 Complex Convolutions 181

3.1 Linear Polynomial Convolutions 181

3.2 Linear Non-Polynomial Convolutions 204

3.2.1 Picard, Poisson-Cauchy and Gauss-Weierstrass Complex Convolutions 205

3.2.2 Complex q-Picard and q-Gauss-Weierstrass Singular Integrals 257

3.2.3 Post-Widder Complex Convolution 264

3.2.4 Rotation-Invariant Complex Convolutions 269

3.2.5 Sikkema Complex Convolutions 279

3.3 Nonlinear Complex Convolutions 286

3.4 Bibliographical Notes and Open Problems 294

4 Appendix: Related Topics 295

4.1 Bernstein Polynomials of Quaternion Variable 295

4.2 Approximation of Vector-Valued Functions 299

4.2.1 Real Variable Case 300

4.2.2 Complex Variable Case 314

4.3 Strong Approximation by Complex Taylor Series 321

4.4 BibliographicalNotes and Open Problems 324

Bibliography 327

Index 337