Approximation by Complex Bernstein and Convolution Type Operators Book

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