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.
Author(s): Sorin G. Gal
Series: Concrete and Applicable Mathematics
Publisher: WS
Year: 2009
Language: English
Pages: 350
Preface......Page 8
Contents......Page 12
1.0 Auxiliary Results in Complex Analysis......Page 14
1.1 Bernstein Polynomials......Page 19
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-Szasz-Mirakjan Operators......Page 116
1.9 Baskakov Operators......Page 137
1.10 Balazs-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-Szasz-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
3.3 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
4.3 Strong Approximation by Complex Taylor Series......Page 334
4.4 Bibliographical Notes and Open Problems......Page 337
Bibliography......Page 340
Index......Page 350
