Approximation By Complex Bernstein And Convolution Type Operators

By: Sorin G Gal

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Hardcover | 12 August 2009

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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-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 (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.

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Preface	p. vii
Bernstein-Type Operators of One Complex Variable	p. 1
Auxiliary Results in Complex Analysis	p. 1
Bernstein Polynomials	p. 6
Bernstein Polynomials on Compact Disks	p. 6
Bernstein-Faber Polynomials on Compact Sets	p. 19
Iterates of Bernstein Polynomials	p. 26
Generalized Voronovskaja Theorems for Bernstein Polynomials	p. 35
Butzer's Linear Combination of Bernstein Polynomials	p. 42
q-Bernstein Polynomials	p. 50
Bernstein-Stancu Polynomials	p. 67
Bernstein-Kantorovich Type Polynomials	p. 96
Favard-Szasz-Mirakjan Operators	p. 103
Baskakov Operators	p. 124
Balazs-Szabados Operators	p. 139
Bibliographical Notes and Open Problems	p. 149
Bernstein-Type Operators of Several Complex Variables	p. 155
Introduction	p. 155
Bernstein Polynomials	p. 156
Favard-Szasz-Mirakjan Operators	p. 166
Baskakov Operators	p. 172
Bibliographical Notes and Open Problems	p. 179
Complex Convolutions	p. 181
Linear Polynomial Convolutions	p. 181
Linear Non-Polynomial Convolutions	p. 204
Picard, Poisson-Cauchy and Gauss-Weierstrass Complex Convolutions	p. 205
Complex q-Picard and q-Gauss-Weierstrass Singular Integrals	p. 257
Post-Widder Complex Convolution	p. 264
Rotation-Invariant Complex Convolutions	p. 269
Sikkema Complex Convolutions	p. 279
Nonlinear Complex Convolutions	p. 286
Bibliographical Notes and Open Problems	p. 294
Appendix: Related Topics	p. 295
Bernstein Polynomials of Quaternion Variable	p. 295
Approximation of Vector-Valued Functions	p. 299
Real Variable Case	p. 300
Complex Variable Case	p. 314
Strong Approximation by Complex Taylor Series	p. 321
Bibliographical Notes and Open Problems	p. 324
Bibliography	p. 327
Index	p. 337
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Approximation By Complex Bernstein And Convolution Type Operators

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