Contents
Preface
Chapter 1 Overview 1
1.1 Introduction 1
1.2 Symbols and Definitions 9
1.3 Sets of Monotone Sequence and Various Generalizations 10
1.3.1 Definitions 10
1.3.2 History and Development 14
1.3.3 Relationships among Sets of Sequences 16
1.4 Notes and Exercises 23
1.4.1 Notes 23
1.4.2 Exercises 25
Chapter 2 Uniform Convergence of Trigonometric Series 26
2.1 Classic Theorems 26
2.2 Development: MVBV Concept in Positive Sense 33
2.3 Further Discussion: In Positive Sense 41
2.4 Breakthrough: MVBV Concept in Real Sense 46
2.5 Notes and Exercises 52
2.5.1 Notes 52
2.5.2 Exercises 53
Chapter 3 L1-Convergence of Fourier Series 55
3.1 History and Development 55
3.2 Further Development: In Positive Sense 66
3.3 Mean Value Bounded Variation: In Real Sense 77
3.4 L1-Approximation 81
3.5 Convexity of Coefficients 89
3.6 Notes and Exercises 93
3.6.1 Notes 93
3.6.2 Exercises 94
Chapter 4 Lp-Integrability of Trigonometric Series 96
4.1 Lp-Integrability 96
4.2 Lp-Convergence 105
4.3 Lp-Integrability for Derivatives 114
4.4 A Conjecture 119
4.5 Notes and Exercises 120
4.5.1 Notes 120
4.5.2 Exercises 121
Chapter 5 Fourier Coefficients and Best Approximation 123
5.1 Classical Results 123
5.2 A Generalization to Strong Mean Value Bounded Variation 124
5.3 Approximation by Fourier Sums with Strong Monotone Coefficients 138
5.3.1 Strong Monotonicity and Fourier Approximation 138
5.3.2 Quasi-Geometric Monotone Conditions 145
5.4 Notes and Exercises 150
5.4.1 Notes 150
5.4.2 Exercises 151
Chapter 6 Integrability of Trigonometric Series 152
6.1 Weighted Integrability: In Positive Sense 152
6.2 Weighted Integrability: In Real Sense 157
6.3 Integrability of Sine Series and Logarithm Bounded Variation Conditions 167
6.4 Logarithm Bounded Variation Conditions: In Real Sense 181
6.5 Integrability of Derivatives 186
6.6 Notes and Exercises 193
6.6.1 Notes 193
6.6.2 Exercises 193
Chapter 7 Other Classical Results in Analysis 194
7.1 Important Trigonometric Inequalities 194
7.2 An Asymptotic Equality 203
7.3 Strong Approximation and Related Embedding Theorems 218
7.4 Abel’s and Dirichlet’s Criteria 227
7.5 Notes and Exercises 231
7.5.1 Notes 231
7.5.2 Exercises 232
Chapter 8 Trigonometric Series with General Coefficients 234
8.1 Piecewise Bounded Variation Conditions 234
8.1.1 “Rarely Changing” Concept 234
8.1.2 Piecewise Bounded Variation 235
8.1.3 Piecewise Mean Value Bounded Variation 236
8.2 No More Piecewise 240
8.3 Notes 241
References 242
Index 249
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