Degree of Approximation By Polynomials in The Complex DomainPDF电子书下载

Chapter Ⅰ．PRELIMINARIES 1

1．Statement of the Problems;Notation 1

2．Lipschitz Conditions and Derivatives 8

3．Exercises 30

4．Discussion 31

PART Ⅰ．PROBLEM α 33

Chapter Ⅱ．POLYNOMIAL INEQUALITIES 33

1．A Polynomial and its Derivarives 33

2．A Polynomial and its Integral 40

3．The Lagrange-Hermite Interpolation Formula 43

4．Geometric Properties of Equipotential Curves 44

5．Equally Distributed Points 48

6．Exercises 75

7．Open Problems 78

Chapter Ⅲ．TCHEBYCHEFF APPROXIMATION 81

1．Jackson＇s Theorem on Trigonometric Approximation 81

2．Direct Theorems 90

3．The Faber Polynomials 96

4．Indirect Theorems 100

5．The Derivative and Integral of a Function 108

6．Best Approximation 111

7．Exercises 112

8．Open Problems 114

Chapter Ⅳ．APPROXIMATION MEASURED BY A LINE INTEGRAL 117

1．Direct Theorems 117

2．Indirect Theorems 120

3．Orthogonal Polynomials 126

4．Polynomials of Best Approximation 131

5．Generality of the Weight Function 133

6．Exercises 135

7．Discussion 139

PART Ⅱ．PROBLEM β 141

Chapter Ⅴ．PRELIMINARIES 141

1．An Interpolation Formula and Inequalities 141

2．An Extended Classification 144

3．A Polynomial and its Integral 149

4．Exercises 152

5．Open Problems 154

Chapter Ⅵ．TCHEBYCHEFF APPROXIMATION 156

1．Direct Theorems 156

2．Operations with Approximating Sequences 158

3．Indirect Theorems 166

4．Counter Examples 170

5．Functions with Isolated Singularities 172

6．Polynomials of Best Approximation 176

7．Exercises 179

8．Discussion 185

Chapter Ⅶ．APPROXIMATION MEASURED BY A LINE INTEGRAL 189

1．Direct Theorems 189

2．Indirect Theorems 190

3．Orthogonal Polynomials 192

4．Polynomials of Best Approximation 194

5．Generality of the Weight Function 195

6．Exercises 196

7．Discussion 198

Chapter Ⅷ．SPECIAL CONFIGURATIONS 200

1．Approximation on |z| = 1 by Polynomials in z and 1/z 200

2．Approximation on -1 ？ z ？ 1 208

3．Trigonometric Approximation 212

4．Direct Methods on Problem α and Problem β 214

5．Exercises 218

6．Discussion 224

Bibliography 226