《Degree of Approximation By Polynomials in The Complex Domain》

Chapter Ⅰ．PRELIMINARIES1

1．Statement of the Problems;Notation1

2．Lipschitz Conditions and Derivatives8

3．Exercises30

4．Discussion31

PART Ⅰ．PROBLEM α33

Chapter Ⅱ．POLYNOMIAL INEQUALITIES33

1．A Polynomial and its Derivarives33

2．A Polynomial and its Integral40

3．The Lagrange-Hermite Interpolation Formula43

4．Geometric Properties of Equipotential Curves44

5．Equally Distributed Points48

6．Exercises75

7．Open Problems78

Chapter Ⅲ．TCHEBYCHEFF APPROXIMATION81

1．Jackson＇s Theorem on Trigonometric Approximation81

2．Direct Theorems90

3．The Faber Polynomials96

4．Indirect Theorems100

5．The Derivative and Integral of a Function108

6．Best Approximation111

7．Exercises112

8．Open Problems114

Chapter Ⅳ．APPROXIMATION MEASURED BY A LINE INTEGRAL117

1．Direct Theorems117

2．Indirect Theorems120

3．Orthogonal Polynomials126

4．Polynomials of Best Approximation131

5．Generality of the Weight Function133

6．Exercises135

7．Discussion139

PART Ⅱ．PROBLEM β141

Chapter Ⅴ．PRELIMINARIES141

1．An Interpolation Formula and Inequalities141

2．An Extended Classification144

3．A Polynomial and its Integral149

4．Exercises152

5．Open Problems154

Chapter Ⅵ．TCHEBYCHEFF APPROXIMATION156

1．Direct Theorems156

2．Operations with Approximating Sequences158

3．Indirect Theorems166

4．Counter Examples170

5．Functions with Isolated Singularities172

6．Polynomials of Best Approximation176

7．Exercises179

8．Discussion185

Chapter Ⅶ．APPROXIMATION MEASURED BY A LINE INTEGRAL189

1．Direct Theorems189

2．Indirect Theorems190

3．Orthogonal Polynomials192

4．Polynomials of Best Approximation194

5．Generality of the Weight Function195

6．Exercises196

7．Discussion198

Chapter Ⅷ．SPECIAL CONFIGURATIONS200

1．Approximation on |z| = 1 by Polynomials in z and 1/z200

2．Approximation on -1 ？ z ？ 1208

3．Trigonometric Approximation212

4．Direct Methods on Problem α and Problem β214

5．Exercises218

6．Discussion224

Bibliography226