Fourier analysis: part 1 Theory
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
[2016]
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| Series: | London Mathematical Society student texts
85 |
| Subjects: | |
| Online Access: | Klappentext Inhaltsverzeichnis |
| Physical Description: | xiv, 353 Seiten |
| ISBN: | 9781107044104 9781107620353 |
Staff View
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Record in the Search Index
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| adam_text | Fourier analysis aims to decompose functions into a superposition of simple
trigonometric functions whose special features can be exploited to isolate
specific components into manageable clusters before reassembling the
pieces.
This two-volume text presents a largely self-contained treatment, compris-
ing not just the major theoretical aspects (Part I) but also exploring links to
other areas of mathematics and applications to science and
technology (Part II). Following the historical and conceptual genesis, this
book (Part I) provides overviews of basic measure theory and functional analy-
sis, with added insight into complex analysis and the theory of distributions.
The material is intended for both beginning and advanced graduate students
with a thorough knowledge of advanced calculus and linear algebra. Historical
notes are provided and topics are illustrated at every stage by examples
and exercises, with separate hints and solutions, thus making the exposition
useful both as a course textbook and for individual study.
Contents
R
Preface page xi
Introduction 1
1.1 Trigonometric polynomials and series 1
1.2 The dawn of the theory 4
1.2.1 The vibrating string controversy 5
1.2.2 Fourier’s view on heat flow 11
1.3 Application: irrationality of n 12
1.4 Exercises 13
1.4.1 Statements 13
1.4.2 Hints 14
1.4.3 Solutions 14
The Lebesgue measure and integral 15
2.1 Historical considerations 15
2.1.1 Ancient measure theory 15
2.1.2 The Riemann integral 18
2.1.3 Outer measure and measurability 22
2.2 A brief outline of the Lebesgue integral 28
2.2.1 Definition and basic properties 28
2.2.2 Multiple integrals 35
2.2.3 The anti-derivative problem 38
Vll
43
47
51
51
57
59
68
71
71
73
83
93
94
99
108
112
124
127
128
137
140
158
159
163
165
176
177
182
182
Contents
2.2.4 Length of curves
2.3 Abstract measure theory
2.4 Exercises
2.4.1 Statements
2.4.2 Hints
2.4.3 Solutions
2.5 Notes to Chapter 2
Elements of functional analysis
3.1 An overall perspective
3.2 Hilbert spaces
3.3 Banach spaces
3.4 Functionals and operators
3.4.1 The family of bounded linear operators
3.4.2 The Hahn—Banach theorems
3.4.3 Baire category and consequences
3.4.4 The spectral theorem
3.5 Frechet spaces
3.6 Exercises
3.6.1 Statements
3.6.2 Hints
3.6.3 Solutions
3.7 Notes to Chapter 3
Convergence results for Fourier series
4.1 Basic properties of Fourier coefficients
4.2 Pointwise convergence
4.3 Mean-square convergence
4.4 Convergence at a jump discontinuity
4.5 Exercises
4.5.1 Statements
Contents ix
4.5.2 Hints 186
4.5.3 Solutions 187
4.6 Notes to Chapter 4 197
5 Fourier transforms 199
5.1 Rapidly decreasing smooth functions 201
5.2 Fourier transform for square integrable functions 205
5.3 Fourier transform for integrable functions 206
5.4 Exercises 208
5.4.1 Statements 208
5.4.2 Hints 210
5.4.3 Solutions 211
5.5 Note to Chapter 5 216
6 Multi-dimensional Fourier analysis 217
6.1 Fourier transform 217
6.1.1 The class of tempered distributions 220
6.1.2 Fourier transform of a tempered distribution 227
6.2 Fourier series 231
6.2.1 L2-convergence 232
6.2.2 Pointwise convergence 232
6.2.3 The tempered distributions approach 235
6.3 Fourier transform of a measure 241
6.4 The Fourier transform on LP(R)~spaces 246
6.5 Sobolev spaces 247
6.6 Periodic Sobolev spaces 254
6.7 Exercises 254
6.7.1 Statements 254
6.7.2 Hints 258
6.7.3 Solutions 260
6.8 Notes to Chapter 6 271
X
Contents
7 A glance at some advanced topics 273
7.1 Complex analysis techniques 273
7.1.1 Basic facts about analytic functions 273
7.1.2 Fourier series convergence by change of variables 31 1
7.1.3 Paley-Wiener theorems 312
7.1.4 Hardy spaces 317
7.2 Pseudodifferential operators 329
Afterword 335
Appendix Historical notes 337
References 343
Index 349
|
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| isbn | 9781107044104 9781107620353 |
| language | English |
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| physical | xiv, 353 Seiten |
| publishDate | 2016 |
| publishDateSearch | 2016 |
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| publisher | Cambridge University Press |
| record_format | marc |
| series | London Mathematical Society student texts |
| series2 | London Mathematical Society student texts |
| spelling | Constantin, Adrian 1970- (DE-588)1020343125 aut Fourier analysis part 1 Theory Adrian Constantin, Universität Wien, Austria Cambridge Cambridge University Press [2016] xiv, 353 Seiten txt rdacontent n rdamedia nc rdacarrier London Mathematical Society student texts 85 London Mathematical Society student texts Harmonische Analyse (DE-588)4023453-8 gnd rswk-swf Harmonische Analyse (DE-588)4023453-8 s DE-604 (DE-604)BV044341046 1 London Mathematical Society student texts 85 (DE-604)BV000841726 85 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029744105&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029744105&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Constantin, Adrian 1970- Fourier analysis London Mathematical Society student texts Harmonische Analyse (DE-588)4023453-8 gnd |
| subject_GND | (DE-588)4023453-8 |
| title | Fourier analysis |
| title_auth | Fourier analysis |
| title_exact_search | Fourier analysis |
| title_full | Fourier analysis part 1 Theory Adrian Constantin, Universität Wien, Austria |
| title_fullStr | Fourier analysis part 1 Theory Adrian Constantin, Universität Wien, Austria |
| title_full_unstemmed | Fourier analysis part 1 Theory Adrian Constantin, Universität Wien, Austria |
| title_short | Fourier analysis |
| title_sort | fourier analysis theory |
| topic | Harmonische Analyse (DE-588)4023453-8 gnd |
| topic_facet | Harmonische Analyse |
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