Some random series of functions:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
[1985]
|
| Edition: | Second edition |
| Series: | Cambridge studies in advanced mathematics
5 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 290 - 300 |
| Physical Description: | xiii, 305 Seiten |
| ISBN: | 052124966X 0521456029 |
Staff View
MARC
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| 100 | 1 | |a Kahane, Jean-Pierre |d 1926-2017 |0 (DE-588)131512579 |4 aut | |
| 245 | 1 | 0 | |a Some random series of functions |c Jean-Pierre Kahane |
| 250 | |a Second edition | ||
| 264 | 1 | |a Cambridge |b Cambridge University Press |c [1985] | |
| 264 | 4 | |c © 1985 | |
| 300 | |a xiii, 305 Seiten | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Cambridge studies in advanced mathematics |v 5 | |
| 500 | |a Literaturverz. S. 290 - 300 | ||
| 650 | 4 | |a Fonctions (Mathématiques) | |
| 650 | 4 | |a Processus stochastiques | |
| 650 | 4 | |a Séries (Mathématiques) | |
| 650 | 4 | |a Variables aléatoires | |
| 650 | 4 | |a Functions | |
| 650 | 4 | |a Random variables | |
| 650 | 4 | |a Series | |
| 650 | 4 | |a Stochastic processes | |
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Record in the Search Index
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Contents
Preface xi
1 A few tools from probability tfaeory 1
1 Introduction 1
2 The basic notions 2
3 Distribution and similarity 3
4 Product probability space 4
5 The Standard model; independence; Steinhaus and
Rademacher sequences 4
6 Integration: the main tools 5
7 Symmetrie random vectors 8
8 Random funetions and analytic sets 9
2 Random series in a Banach space 11
1 Introduction 11
2 Summability methods 12
3 Sums of Symmetrie random vectors; two lemmas 14
4 Proof of theorem 1 15
5 Rademacher series £f + u„ 18
6 A principle of contraction 20
7 The strong integrability for Rademacher series 23
8 Exercises 25
3 Random series in a Hubert space 28
1 Introduction 28
2 The Kolmogorov inequality 29
3 The Paley Zygmund inequalities 30
4 Positive random series 32
v
vi Contents ,
5 Necessary and sufficient conditions for convergence
and boundedness 33
6 Exercises 35
4 Random Taylor series 37
1 Introduction 37 ,
2 Singular points 38
3 The symmetric case 39
4 The general case 40
5 Random Taylor series in two complex variables 41
6 Random Dirichlet series 43
7 Complements and exercises 44
5 Random Fourier series 46
1 Introduction 46
2 Auxiliary results on trigonometric series 47
3 Rademacher series: the case Yj? x^ = oo 49
4 Rademacher series: the case Y,o x* °° 51
5 The general Paley Zygmund theorem 53
6 Auxiliary results on series of translates 54
7 Convergence and boundedness in C or L* 56
8 Convergence everywhere; the Billard theorem 58
9 An application: Fourier coefficients of continuous
functions 60
10 Exercises 63
6 A bound for random trigonometric polynomials
and applications 67
1 Introduction 67
2 Distribution of M= || P ^ 68
3 Applications; a theorem of Littlewood and Salem;
Sidon and Helson sets 70
4 Another application: generalized almost periodic
sequences 72
5 Polynomials with unimodular coefficients 75
6 Sums of sinuses 78
7 Exercises 81
7 Conditions on coefficients for regularity S3
1 Introduction 83
2 A sufficient condition for (l)e C 84
3 Estimates for the modulus of continuity (subgaussian
case) 86
Contents vii
4 A sufficient condition for (1) e Aa 88
5 An application 90
6 Exercises 91
8 Conditions on coefficients for irregularity 93
1 Introduction 93
2 Unboundedness: the Paley Zygmund approach 94
3 Unboundedness: a particular case 96
4 Unboundedness: the general case 98
5 Irregularity almost everywhere 99
6 Irregularity everywhere 101
7 Simultaneous inequalities 103
8 Irregularity everywhere (continued) 104
9 Divergence everywhere 106
10 Exercises 108
9 Random point masses on the circle 109
1 Introduction 109
2 Two theorems on Fourier Stieltjes series 110
3 Proof of theorem 2 112
4 An almost everywhere divergent Fourier series 116
5 Poisson transform of £" £j/n, 59j. 118
6 A theorem on conjugate harmonic functions 121
7 More about the case £? m) = 1 124
8 Exercises 125
10 A few geometric notions 128
1 Introduction 128
2 Hausdorff measures and dimensions; Frostman's lemma 129
3 Energy and capacity; Frostman's theorem 132
4 e covering numbers 134
5 Helices 135
6 Quasi helices; von Koch and Assouad curves 137
7 More on dimensions 139
8 Exercises 141
11 Random translates and covering 143
1 Introduction 143
2 Covering the circle: a sufficient condition 144
3 Covering the circle: a necessary condition 149
4 Covering the circle: the necessary and sufficient
condition 150
viii Contents I
5 Covering a subset of T* by random sets: a necessary
153
6 Covering a subset of Jq: a sufficient condition; the case
of convex gn 156
7 The case of non flattening convex gn; covering a set
of given Hausdorff dimension 158
8 The case of non flattening convex gn (continued);
dimension of the non covered set 159
9 Concluding remarks 161
10 Exercises 162
12 Gaussian variables and gaussian series 165
1 Introduction 165
2 Formulas on Fourier transforms 166
3 Gaussian random variables 168
4 Some more formulas 171
5 Around the Borel Cantelli lemma 172
6 Transient and recurrent gaussian series 173
7 Gaussian series in a Banach space 175
8 Exercises 177
13 Gaussian Taylor series 178
1 Introduction 178
2 A review of previous results 179
3 The range of F(z)(|z| l) 180
4 The radial behavior: a recurrence condition 184
5 The radial behavior: transience conditions 186
6 Non radial behavior: recurrence conditions 189
7 Transience on circular sets 193
8 Exercises 195
14 Gaussian Fourier series 197
1 Introduction 197
2 Review of known results 199
3 Capacities and Hausdorff dimension reviewed 199
4 Range of F 200
5 The zeros of F 203
6 A definition of diq\F) 207
7 The Malliavin theorem on spectral synthesis 209 ,
8 Exercises 210
Contents ix
15 Boundedness and continuity for gaussian processes 211
1 Introduction 211
2 Slepian's lemma 213
3 Marcus and Shepp's theorem; the Pisier algebra 215
4 Dudley's theorem 218
5 Fernique's theorem 221
6 Non gaussian Fourier series 226
7 Exercises 231
16 The brownian motion 233
1 Introduction 233
2 The Wiener process 233
3 The Fourier Wiener series 235
4 More on local properties 237
5 Stopping times, polar sets and newtonian capacity 242
6 Self crossing 245
17 Brownian images in harmonic analysis 250
1 Introduction 250
2 Brownian images 251
3 Brownian image of a measure; proof of theorem 1 253
4 Arithmetical properties of brownian images; proof of
theorem 2 255
5 Image of a measure by a gaussian Fourier series 257
6 A construction of H. Cartan; proof of lemma 6 258
7 A generalization of theorems 1 and 2 260
8 Exercises 261
18 Fractional brownian images and level sets 263
1 Introduction 263
2 The gaussian processes (n, d, y) 264
3 Fractional brownian image of a measure; new Salem sets 265
4 Fractional brownian images (continued); occupation
density 267
5 Level sets 272
6 Uniqueness and continuity of 3(X — x) 275
7 Graphs 278
8 Exercises 279
Notes 281
Bibliography 290
Index 301 |
| any_adam_object | 1 |
| author | Kahane, Jean-Pierre 1926-2017 |
| author_GND | (DE-588)131512579 |
| author_facet | Kahane, Jean-Pierre 1926-2017 |
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| dewey-full | 519.2 515 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics 515 - Analysis |
| dewey-raw | 519.2 515 |
| dewey-search | 519.2 515 |
| dewey-sort | 3519.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Second edition |
| format | Book |
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| indexdate | 2024-09-04T00:07:17Z |
| institution | BVB |
| isbn | 052124966X 0521456029 |
| language | English |
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| physical | xiii, 305 Seiten |
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| series2 | Cambridge studies in advanced mathematics |
| spelling | Kahane, Jean-Pierre 1926-2017 (DE-588)131512579 aut Some random series of functions Jean-Pierre Kahane Second edition Cambridge Cambridge University Press [1985] © 1985 xiii, 305 Seiten txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 5 Literaturverz. S. 290 - 300 Fonctions (Mathématiques) Processus stochastiques Séries (Mathématiques) Variables aléatoires Functions Random variables Series Stochastic processes Funktionenreihe (DE-588)4155689-6 gnd rswk-swf Zufallsvariable (DE-588)4129514-6 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Funktionenreihe (DE-588)4155689-6 s Zufallsvariable (DE-588)4129514-6 s DE-604 Stochastischer Prozess (DE-588)4057630-9 s 1\p DE-604 Cambridge studies in advanced mathematics 5 (DE-604)BV000003678 5 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001262551&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Kahane, Jean-Pierre 1926-2017 Some random series of functions Cambridge studies in advanced mathematics Fonctions (Mathématiques) Processus stochastiques Séries (Mathématiques) Variables aléatoires Functions Random variables Series Stochastic processes Funktionenreihe (DE-588)4155689-6 gnd Zufallsvariable (DE-588)4129514-6 gnd Stochastischer Prozess (DE-588)4057630-9 gnd |
| subject_GND | (DE-588)4155689-6 (DE-588)4129514-6 (DE-588)4057630-9 |
| title | Some random series of functions |
| title_auth | Some random series of functions |
| title_exact_search | Some random series of functions |
| title_full | Some random series of functions Jean-Pierre Kahane |
| title_fullStr | Some random series of functions Jean-Pierre Kahane |
| title_full_unstemmed | Some random series of functions Jean-Pierre Kahane |
| title_short | Some random series of functions |
| title_sort | some random series of functions |
| topic | Fonctions (Mathématiques) Processus stochastiques Séries (Mathématiques) Variables aléatoires Functions Random variables Series Stochastic processes Funktionenreihe (DE-588)4155689-6 gnd Zufallsvariable (DE-588)4129514-6 gnd Stochastischer Prozess (DE-588)4057630-9 gnd |
| topic_facet | Fonctions (Mathématiques) Processus stochastiques Séries (Mathématiques) Variables aléatoires Functions Random variables Series Stochastic processes Funktionenreihe Zufallsvariable Stochastischer Prozess |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001262551&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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