Approximation by Complex Bernstein and Convolution Type Operators

✍ Scribed by Sorin G. Gal

• Publisher: World Scientific Publishing Company, Incorporated
• Year: 2009
• Tongue: English
• Leaves: 350
• Series: Series on Concrete and Applicable Mathematics 8
• Category: Library

✦ Synopsis

This 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-Szasz-Mirakjan, Baskakov and Balazs-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 Vallee Poussin, Fejer, 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 (PDE) also are 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.

✦ Table of Contents

Contents......Page 12
Preface......Page 8
1.0 Auxiliary Results in Complex Analysis......Page 14
1.1.1 Bernstein Polynomials on Compact Disks......Page 19
1.1.2 Bernstein-Faber Polynomials on Compact Sets......Page 32
1.2 Iterates of Bernstein Polynomials......Page 39
1.3 Generalized Voronovskaja Theorems for Bernstein Polynomials......Page 48
1.4 Butzer's Linear Combination of Bernstein Polynomials......Page 55
1.5 q-Bernstein Polynomials......Page 63
1.6 Bernstein-Stancu Polynomials......Page 80
1.7 Bernstein-Kantorovich Type Polynomials......Page 109
1.8 Favard-Sz asz-Mirakjan Operators......Page 116
1.9 Baskakov Operators......Page 137
1.10 Bal azs-Szabados Operators......Page 152
1.11 Bibliographical Notes and Open Problems......Page 162
2.1 Introduction......Page 168
2.2 Bernstein Polynomials......Page 169
2.3 Favard-Sz asz-Mirakjan Operators......Page 179
2.4 Baskakov Operators......Page 185
2.5 Bibliographical Notes and Open Problems......Page 192
3.1 Linear Polynomial Convolutions......Page 194
3.2 Linear Non-Polynomial Convolutions......Page 217
3.2.1 Picard, Poisson-Cauchy and Gauss-Weierstrass Complex Convolutions......Page 218
3.2.2 Complex q-Picard and q-Gauss-Weierstrass Singular Integrals......Page 270
3.2.3 Post-Widder Complex Convolution......Page 277
3.2.4 Rotation-Invariant Complex Convolutions......Page 282
3.2.5 Sikkema Complex Convolutions......Page 292
3.3 Nonlinear Complex Convolutions......Page 299
3.4 Bibliographical Notes and Open Problems......Page 307
4.1 Bernstein Polynomials of Quaternion Variable......Page 308
4.2 Approximation of Vector-Valued Functions......Page 312
4.2.1 Real Variable Case......Page 313
4.2.2 Complex Variable Case......Page 327
4.3 Strong Approximation by Complex Taylor Series......Page 334
4.4 Bibliographical Notes and Open Problems......Page 337
Bibliography......Page 340
Index......Page 350