Algebraic Numbers and Harmonic Analysis (eBook)

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2000 | 1. Auflage
273 Seiten
Elsevier Science (Verlag)
978-0-08-095412-7 (ISBN)

Infos

Algebraic numbers and harmonic analysis

Inhalt

Front Cover 1
Algebraic Numbers and Harmonic Analysis 4
Copyright Page 5
Contents 8
Preface 6
List of symbols 11
Introduction 12
Chapter I. Diophantine approximations to real numbers 15
1. Some classical results in Diophantine approximations 15
2. Measure-theoretical methods in Diophantine approximations 18
3. Badly approximable real numbers 21
4. Diophantine approximations and prediction of the size of trigonometric sums 28
5. Kronecker's theorem and problems with a large number of equations 32
6. Pisot and Salem numbers 36
7. Pisot numbers, Salem numbers and harmonious sets of real numbers 46
8. Notes 50
Chapter II. Diophantine approximations and additive problems in locally compact abelian groups 51
1. Preliminaries on l.c.a. groups 51
2. Harmonious sets 53
3. Basic properties of harmonious sets 54
4. Construction of harmonious sets in l.c.a. groups (the lacunary case) 57
5. Construction of relatively dense harmonious sets 59
6. Relatively dense harmonious sets of real numbers 65
7. Quantitative problems 66
8. Harmonious sets of real numbers closed under multiplication 68
9. Another definition of models of real numbers 70
10. Harmonious sets in p-adic fields 71
11. Counter-examples 75
12. Pisot-Salem-Chabauty numbers in Qp 75
13. Adeles and harmonious sets closed under multiplication 79
14. Characterization of harmonious sets by additive properties 82
15. Notes 91
Chapter III. Uniqueness of representation by trigonometric series 92
1. Riemann's theory 92
2. Symmetric sets 98
3. Pisot numbers and uniqueness 100
4. Results for symmetric sets 104
5. Other results for symmetric sets 107
6. Sets of uniqueness in general l.c.a. groups 108
7. The p-adic case 111
8. Generalizations 113
9. Other results on the problem of uniqueness 114
10. Notes 114
Chapter IV. Problems on a-periodic trigonometric sums 115
1. Elementary properties of almost-periodic functions 116
2. Classes of almost-periodic functions 117
3. Dilations and classes of almost-periodic functions on the line 120
4. Classes of almost-periodic functions and a priori estimates on the size 121
5. Pisot numbers and coherent sets of frequencies: the real case 121
6. A set of powers 123
7. Pisot numbers and coherent sets of frequencies: the p-adic case 123
8. The groups C and R2 124
9. Coherent sets of frequencies in l.c.a, groups 125
10. Coherent sets of frequencies and restriction algebras 126
11. Other definitions of coherent sets of frequencies 130
12. Other relations between A(E), B(E) and L8e 131
13. Slowly-perturbed trigonometric sums 132
14. Coherent sets of frequencies and isomorphisms 137
15. Improvement on the preceding results when . is harmonious 139
16. A new characterization of Pisot numbers 147
17. Slowly-perturbed trigonometric sums in the real case 149
18. Notes 150
Interlude 151
Chapter V. Special trigonometric series (complex methods) 153
1. The Laplace transform 153
2. The Paley-Wiener theorem 157
3. Repartition of roots of entire function of exponential type 159
4. Bernstein's inequality 160
5. Other inequalities of Bernstein type 161
6. Irrational numbers and special trigonometric series 163
7. Special trigonometric series: the a-periodic case 168
8. Special trigonometric series on p-adic fields 173
9. Mean periodic functions 180
10. Notes 192
Chapter VI. Special trigonometric series (group-theoretic methods) 194
A. Topological Sidon sets of real numbers 194
1. Definition and basic properties of topological Sidon sets 194
2. Examples of topological Sidon sets 196
3. Construction of remarkable measures associated with topological Sidon sets 196
4. The union of two topological Sidon sets 200
5. Topological Sidon sets and stability of coherent sets of frequencies 201
6. Interpolation of bounded functions defined on a topological Sidon set by the Fourier-transforms of complex bounded Radon measures on R 204
7. Estimates for trigonometric sums whose frequencies belong to a topological Sidon set 205
B. Idempotent and semi-idempotent measures 210
8. Idempotent measures on l.c.a. groups 210
9. Semi-idempotent measures 217
10. Behaviour at infinity of special a-periodic trigonometric series 222
11. Notes 223
Chapter VII. Pisot numbers and spectral synthesis 224
1. Spectral synthesis and structure of closed ideals of a group algebra 224
2. Spectral synthesis and atomization of distributions 228
3. A strong form of spectral synthesis 229
4. Spectral synthesis and weighted approximation 232
5. Pisot numbers and spectral synthesis 233
6. Bochner's property for a Banach algebra 237
7. Bochner's property and harmonic analysis 239
8. The p-adic case 244
9. Notes 248
Chapter VIII. Ultra-thin symmetric sets 249
1. Properties of ultra-thin symmetric sets 249
2. Functions whose spectra lie in ultra-thin symmetric sets 250
3. Theorem I as a corollary of Theorem II 253
4. Proof of Theorem II: reduction to a problem on intervals 255
5. Symmetric coherent sets of frequencies 258
6. High frequencies and independence complements
7. From the group R to the group D8 262
8. Spectral synthesis fails in A(D8) 264
9. Spectral synthesis fails in A(E) and A(R) 266
10. Counter-examples. A random set due to Salem 267
11. Other random sets 271
12. Notes 273
Conclusion 274
Some open problems about symmetric sets 275
Open problems. Special trigonometric series on local fields 276
Appendix 277
Bibliography 281
Index 285

Produkt-Details

• Erscheint lt. Verlag: 1.4.2000
• Mitarbeit: Herausgeber (Serie): Yves Meyer
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Naturwissenschaften
• Themenwelt: Sozialwissenschaften
• Themenwelt: Technik
• ISBN-10: 0-08-095412-X / 008095412X
• ISBN-13: 978-0-08-095412-7 / 9780080954127