《三角级数收敛中的单调性条件（英文版）》旨在对三角（或Fourier）级数系数单调性条件的设置进行研究，以保证级数的各种收敛性。在对其历史和发展进行了系统回顾的基础上，《三角级数收敛中的单调性条件（英文版）》重点关注近期新的研究进展：对系数的设置既包含单调性的最终推广，同时在此框架下取消原有的正性，力求内容的系统性和原创性，而在论述证明过程中包含了新的思想、方法和技术。可为感兴趣的数学工作者和学生提供相关信息和更新素材。《三角级数收敛中的单调性条件（英文版）》内容自洽，读者只需要具备一定分析知识便可阅读，并可选作研究生教材。

Preface
Acknowledgements
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