6.6 Notes and Exercises

Kapitel in diesem Buch

1. Frontmatter i
2. Preface iii
3. Acknowledgements v
4. Contents vii
5. Chapter 1 Overview
6. 1.1 Introduction 1
7. 1.2 Symbols and Definitions 9
8. 1.3 Sets of Monotone Sequence and Various Generalizations 10
9. 1.4 Notes and Exercises 23
10. Chapter 2 Uniform Convergence of Trigonometric Series
11. 2.1 Classic Theorems 26
12. 2.2 Development: MVBV Concept in Positive Sense 33
13. 2.3 Further Discussion: In Positive Sense 41
14. 2.4 Breakthrough: MVBV Concept in Real Sense 46
15. 2.5 Notes and Exercises 52
16. Chapter 3 L¹-Convergence of Fourier Series
17. 3.1 History and Development 55
18. 3.2 Further Development: In Positive Sense 66
19. 3.3 Mean Value Bounded Variation: In Real Sense 77
20. 3.4 L¹-Approximation 81
21. 3.5 Convexity of Coefficients 89
22. 3.6 Notes and Exercises 93
23. Chapter 4 Lp-Integrability of Trigonometric Series
24. 4.1 L^p-Integrability 96
25. 4.2 L^p-Convergence 105
26. 4.3 L^p-Integrability for Derivatives 114
27. 4.4 A Conjecture 119
28. 4.5 Notes and Exercises 120
29. Chapter 5 Fourier Coefficients and Best Approximation
30. 5.1 Classical Results 123
31. 5.2 A Generalization to Strong Mean Value Bounded Variation 124
32. 5.3 Approximation by Fourier Sums with Strong Monotone Coefficients 138
33. 5.4 Notes and Exercises 150
34. Chapter 6 Integrability of Trigonometric Series
35. 6.1 Weighted Integrability: In Positive Sense 152
36. 6.2 Weighted Integrability: In Real Sense 157
37. 6.3 Integrability of Sine Series and Logarithm Bounded Variation Conditions 167
38. 6.4 Logarithm Bounded Variation Conditions: In Real Sense 181
39. 6.5 Integrability of Derivatives 186
40. 6.6 Notes and Exercises 193
41. Chapter 7 Other Classical Results in Analysis
42. 7.1 Important Trigonometric Inequalities 194
43. 7.2 An Asymptotic Equality 203
44. 7.3 Strong Approximation and Related Embedding Theorems 218
45. 7.4 Abel’s and Dirichlet’s Criteria 227
46. 7.5 Notes and Exercises 231
47. Chapter 8 Trigonometric Series with General Coefficients
48. 8.1 Piecewise Bounded Variation Conditions 234
49. 8.2 No More Piecewise 240
50. 8.3 Notes 241
51. References 242
52. Index 249