Introduction to fourier analysis and wavelets:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Mathematical Soc.
2009
|
| Series: | Graduate studies in mathematics
102 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Originally published: Pacific Grove, CA : Brooks/Cole, c2002. |
| Physical Description: | XVIII, 376 S. graph. Darst. |
| ISBN: | 9780821847978 |
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| 100 | 1 | |a Pinsky, Mark A. |d 1940- |e Verfasser |0 (DE-588)108438791 |4 aut | |
| 245 | 1 | 0 | |a Introduction to fourier analysis and wavelets |c Mark A. Pinsky |
| 264 | 1 | |a Providence, RI |b American Mathematical Soc. |c 2009 | |
| 300 | |a XVIII, 376 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate studies in mathematics |v 102 | |
| 500 | |a Originally published: Pacific Grove, CA : Brooks/Cole, c2002. | ||
| 650 | 4 | |a Fourier analysis | |
| 650 | 4 | |a Wavelets (Mathematics) | |
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| _version_ | 1804138913638383616 |
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| adam_text | CONTENTS
1
FOURIER
SERIES
ON THE CIRCLE
1
1.1
Motivation and Heuristics
1
1.1.1
Motivation from Physics
1
1.1.1.1
The Vibrating String
1
1.1.1.2
Heat Flow in Solids
2
1.1.2
Absolutely Convergent Trigonometric Series
3
1.1.3
*Examples of Factorial and Bessel Functions
6
1.1.4
Poisson
Kernel Example
7
1.1.5
♦Proof of Laplace s Method
9
1.1.6
*Nonabsolutely Convergent Trigonometric Series
11
1.2
Formulation of Fourier Series
13
1.2.1
Fourier Coefficients and Their Basic Properties
13
1.2.2
Fourier Series of Finite Measures
19
1.2.3
*Rates of Decay of Fourier Coefficients
20
1.2.3.1
Piecewise Smooth Functions
21
1.2.3.2
Fourier Characterization of Analytic Functions
22
1.2.4
Sine Integral 24
1.2.4.1
Other Proofs That Si(oo)
= 1 24
1.2.5
Pointwise Convergence Criteria
25
1.2.6
*Integration of Fourier Series
29
1.2.6.1
Convergence of Fourier Series of Measures
30
1.2.7
Riemann Localization Principle
31
1.2.8
Gibbs-Wilbraham Phenomenon
31
1.2.8.1
The General Case
34
1.3
Fourier Series in L2 35
1.3.1
Mean Square Approximation—
Parsevaľs
Theorem
35
1.3.2
*Application
to the Isoperimetric Inequality
38
ix
X
CONTENTS
1.3.3 *Rates
of Convergence in
Lr
39
1.3.3.1
Application to Absolutely-Convergent Fourier
Series
43
1.4
Norm Convergence and Summability
45
1.4.1
Approximate Identities
45
1.4.1.1
Almost
-Е
very where Convergence of the Abel
Means
49
1.4.2
Summability Matrices
51
1.4.3
Fejér
Means of a Fourier Series
54
1.4.3.1
Wiener s Closure Theorem on the Circle
57
1.4.4
*Equidistribution Modulo One
57
1.4.5
*Hardy s Tauberian Theorem
59
1.5
Improved Trigonometric Approximation
61
1.5.1
Rates of Convergence in C(T)
61
1.5.2
Approximation with
Fejér
Means
62
1.5.3
♦Jackson s Theorem
65
1.5.4
♦Higher-Order Approximation
66
1.5.5
♦Converse Theorems of Bernstein
70
1.6
Divergence of Fourier Series
73
1.6.1
The Example of
du Bois-Reymond 74
1.6.2
Analysis via Lebesgue Constants
75
1.6.3
Divergence in the Space
Ľ
78
1.7 *
Appendix: Complements on Laplace s Method
80
1.7.0.1
First Variation on the Theme-Gaussian
Approximation
80
1.7.0.2
Second Variation on the Theme-Improved Error
Estimate
80
1.7.1 *
Application to Bessel Functions
81
1.7.2
♦The Local Limit Theorem of DeMoivre-Laplace
82
1.8
Appendix: Proof of the Uniform Boundedness Theorem
84
1.9
♦Appendix: Higher-Order Bessel functions
85
1.10
Appendix: Cantor s Uniqueness Theorem
86
2
FOURIER TRANSFORMS ON THE LINE AND SPACE
89
2.1
Motivation and Heuristics
89
2.2
Basic Properties of the Fourier Transform
91
2.2.1
Riemann-Lebesgue Lemma
94
2.2.2
Approximate Identities and Gaussian Summability
97
2.2.2.1
Improved Approximate Identities for Pointwise
Convergence
100
2.2.2.2
Application to the Fourier Transform
102
2.2.2.3
The n-Dimensional
Poisson
Kernel
106
CONTENTS Xl
2.2.3
Fourier
Transforms of Tempered Distributions
108
2.2.4
♦Characterization of the Gaussian Density
109
2.2.5
♦Wiener s Density Theorem
110
2.3
Fourier Inversion in One Dimension
112
2.3.1
Dirichlet Kernel and Symmetric Partial Sums
112
2.3.2
Example of the Indicator Function
114
2.3.3
Gibbs-Wilbraham Phenomenon
115
2.3.4
Dini
Convergence Theorem
115
2.3.4.1
Extension to Fourier s Single Integral
117
2.3.5
Smoothing Operations in
R
-Averaging and Summability
117
2.3.6
Averaging and Weak Convergence
118
2.3.7
Cesara
Summability
119
2.3.7.1
Approximation Properties of the
Fejér
Kernel
121
2.3.8
Bernstein s Inequality
122
2.3.9
♦One-Sided Fourier Integral Representation
124
2.3.9.1
Fourier Cosine Transform
124
2.3.9.2
Fourier Sine Transform
125
2.3.9.3
Generalized /¡-Transform
125
2.4
L2 Theory in R
128
2.4.1
Plancherel s Theorem
128
2.4.2
♦Bernstein s Theorem for Fourier Transforms
129
2.4.3
The Uncertainty Principle
131
2.4.3.1
Uncertainty Principle on the Circle
133
2.4.4
Spectral Analysis of the Fourier Transform
134
2.4.4.1
Hermite Polynomials
134
2.4.4.2
Eigenfunction of the Fourier Transform
136
2.4.4.3
Orthogonality Properties
137
2.4.4.4
Completeness
138
2.5
Spherical Fourier Inversion in
Ш
139
2.5.1
Bochner s Approach
139
2.5.2
Piecewise Smooth Viewpoint
145
2.5.3
Relations with the Wave Equation
146
2.5.3.1
The Method of Brandolini and Colzani
149
2.5.4
Bochner-Riesz Summability
152
2.5.4.1
A General Theorem on Almost-Everywhere
Summability
153
2.6
Bessel Functions
154
2.6.1
Fourier Transforms of Radial Functions
157
2.6.2
Lr-Restriction Theorems for the Fourier Transform
158
2.6.2.1
An Improved Result
159
2.6.2.2
Limitations on the Range of
p
161
2.7
The Method of Stationary Phase
162
2.7.1
Statement of the Result
163
2.7.2
Application to Bessel Functions
164
2.7.3
Proof of the Method of Stationary Phase
165
2.7.4
Abel s Lemma
167
Xli CONTENTS
3
FOURIER
ANALYSIS IN IP SPACES
169
3.1
Motivation and Heuristics
169
3.2
The M. Riesz-Thorin Interpolation Theorem
169
3.2.0.1
Generalized Young s Inequality
174
3.2.0.2
The Hausdorff-Young Inequality
174
3.2.1
Stein s Complex Interpolation Theorem
175
3.3
The Conjugate Function or Discrete Hubert Transform
176
3.3.1
U
Theory of the Conjugate Function
177
3.3.2
L1 Theory of the Conjugate Function
179
3.3.2.1
Identification as a Singular Integral
183
3.4
The Hilbert Transform on
R
184
3.4.1
L2
Theory of the Hilbert Transform
185
3.4.2
U
Theory of the Hilbert Transform,
1 <
p
<
oo
186
3.4.2.1
Applications to Convergence of Fourier Integrals
187
3.4.3
Ľ
Theory of the Hilbert Transform and Extensions
188
3.4.3.1
Kolmogorov s Inequality for the Hilbert
Transform
192
3.4.4
Application to Singular Integrals with Odd Kernels
194
3.5
Hardy-Littlewood Maximal Function
197
3.5.1
Application to the Lebesgue Differentiation Theorem
200
3.5.2
Application to Radial Convolution Operators
202
3.5.3
Maximal Inequalities for Spherical Averages
203
3.6
The Marcinkiewicz Interpolation Theorem
206
3.7
Calderón-Zygmund
Decomposition
209
3.8
A Class of Singular Integrals
210
3.9
Properties of Harmonic Functions
212
3.9.1
General Properties
212
3.9.2
Representation Theorems in the Disk
214
3.9.3
Representation Theorems in the Upper Half-Plane
216
3.9.4
Herglotz/Bochner Theorems and Positive Definite
Functions
219
4
POISSON
SUMMATION FORMULA AND
MULTIPLE FOURIER SERIES
222
4.1
Motivation and Heuristics
222
4.2
The
Poisson
Summation Formula in
M
223
4.2.1
Periodization of a Function
223
4.2.2
Statement and Proof
225
4.2.3
Shannon Sampling
228
4.3
Multiple Fourier Series
230
4.3.1
Basic
L
Theory
231
4.3.1.1
Pointwise Convergence for Smooth Functions
233
4.3.1.2
Representation of Spherical Partial Sums
233
CONTENTS
ХІІІ
4.3.2 Basic
L2
Theory
235
4.3.3
Restriction
Theorems
for Fourier Coefficients
236
4.4
Poisson
Summation Formula in Rd
238
4.4.1
♦Simultaneous Nonlocalization
239
4.5
Application to Lattice Points
241
4.5.1
Kendall s Mean Square Error
241
4.5.2
Landau s Asymptotic Formula
243
4.5.3
Application to Multiple Fourier Series
244
4.5.3.1
Three-Dimensional Case
245
4.5.3.2
Higher-Dimensional Case
247
4.6 Schrödinger
Equation and Gauss Sums
247
4.6.1
Distributions on the Circle
248
4.6.2
The
Schrödinger
Equation on the Circle
250
4.7
Recurrence of Random Walk
252
5
APPLICATIONS TO PROBABILITY THEORY
256
5.1
Motivation and Heuristics
256
5.2
Basic Definitions
256
5.2.1
The Central Limit Theorem
260
5.2.1.1
Restatement in Terms of Independent
Random Variables
261
5.3
Extension to Gap Series
262
5.3.1
Extension to Abel Sums
266
5.4
Weak Convergence of Measures
268
5.4.1
An Improved Continuity Theorem
269
5.4.1.1
Another Proof of Bochner
s
Theorem
270
5.5
Convolution Semigroups
272
5.6
The
Berry-Esséen
Theorem
276
5.6.1
Extension to Different Distributions
279
5.7
The Law of the Iterated Logarithm
280
6
INTRODUCTION TO WAVELETS
284
6.1
Motivation and Heuristics
284
6.1.1
Heuristic Treatment of the Wavelet Transform
285
6.2
Wavelet Transform
286
6.2.0.1
Wavelet Characterization of Smoothness
290
6.3 Haar
Wavelet Expansion
291
6.3.1 Haar
Functions and
Haar
Series
291
6.3.2 Haar
Sums and Dyadic Projections
292
6.3.3
Completeness of the
Haar
Functions
295
6.3.3.1 Haar
Series in Co and Lp Spaces
296
6.3.3.2
Pointwise Convergence of
Haar
Series
298
XIV
CONTENTS
6.3.4
*Constraction of Standard Brownian
Motion
299
6.3.5
*Haar
Function Representation of Brownian Motion
301
6.3.6
*Proof of Continuity
301
6.3.7
♦Levy s Modulus of Continuity
302
6.4
Multiresolution Analysis
303
6.4.1
Orthonormal
Systems and Riesz Systems
304
6.4.2
Scaling Equations and Structure Constants
310
6.4.3
From Scaling Function to
MRA
313
6.4.3.1
Additional Remarks
315
6.4.4
Meyer Wavelets
318
6.4.5
From Scaling Function to
Orthonormal
Wavelet
319
6.4.5.1
Direct Proof that V{
θ
Vo Is Spanned by
{^(t-k)}keZ
324
6.4.5.2
Null Integrability of Wavelets Without
Scaling Functions
325
6.5
Wavelets with Compact Support
326
6.5.1
From Scaling Filter to Scaling Function
327
6.5.2
Explicit Construction of Compact Wavelets
330
6.5.2.1
Daubechies Recipe
331
6.5.2.2
Hernandez-Weiss Recipe
333
6.5.3
Smoothness of Wavelets
334
6.5.3.1
A Negative Result
336
6.5.4
Cohen s Extension of Theorem
6.5.1 338
6.6
Convergence Properties of Wavelet Expansions
341
6.6.1
Wavelet Series in L Spaces
341
6.6.1.1
Large Scale Analysis
345
6.6.1.2
Almost
-Е
very where Convergence
346
6.6.1.3
Convergence at a Preassigned Point
347
6.6.2
Jackson and Bernstein Approximation Theorems
347
6.7
Wavelets in Several Variables
352
6.7.1
Two Important Examples
352
6.7.1.1
Tensor Product of Wavelets
354
6.7.2
General Formulation of
MRA
and Wavelets in
ЋУ
354
6.7.2.1
Notations for Subgroups and Cosets
355
6.7.2.2
Riesz Systems and
Orthonormal
Systems in M.d
356
6.7.2.3
Scaling Equation and Structure Constants
357
6.7.2.4
Existence of the Wavelet Set
358
6.7.2.5
Proof That the Wavelet Set Spans V,
θ
Vo
361
6.7.2.6
Cohen s Theorem in Rd
362
6.7.3
Examples of Wavelets in Rd
362
References
365
Notations
369
Index
373
|
| any_adam_object | 1 |
| author | Pinsky, Mark A. 1940- |
| author_GND | (DE-588)108438791 |
| author_facet | Pinsky, Mark A. 1940- |
| author_role | aut |
| author_sort | Pinsky, Mark A. 1940- |
| author_variant | m a p ma map |
| building | Verbundindex |
| bvnumber | BV035459289 |
| callnumber-first | Q - Science |
| callnumber-label | QA403 |
| callnumber-raw | QA403.5 |
| callnumber-search | QA403.5 |
| callnumber-sort | QA 3403.5 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 450 |
| ctrlnum | (OCoLC)500583259 (DE-599)BVBBV035459289 |
| dewey-full | 515.2433 515/.2433 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.2433 515/.2433 |
| dewey-search | 515.2433 515/.2433 |
| dewey-sort | 3515.2433 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| genre | (DE-588)4151278-9 Einführung gnd-content |
| genre_facet | Einführung |
| id | DE-604.BV035459289 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:35:45Z |
| institution | BVB |
| isbn | 9780821847978 |
| language | English |
| lccn | 2008047419 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017379162 |
| oclc_num | 500583259 |
| open_access_boolean | |
| owner | DE-20 DE-634 DE-19 DE-BY-UBM DE-83 DE-29T DE-703 DE-188 |
| owner_facet | DE-20 DE-634 DE-19 DE-BY-UBM DE-83 DE-29T DE-703 DE-188 |
| physical | XVIII, 376 S. graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | American Mathematical Soc. |
| record_format | marc |
| series | Graduate studies in mathematics |
| series2 | Graduate studies in mathematics |
| spelling | Pinsky, Mark A. 1940- Verfasser (DE-588)108438791 aut Introduction to fourier analysis and wavelets Mark A. Pinsky Providence, RI American Mathematical Soc. 2009 XVIII, 376 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 102 Originally published: Pacific Grove, CA : Brooks/Cole, c2002. Fourier analysis Wavelets (Mathematics) Harmonische Analyse (DE-588)4023453-8 gnd rswk-swf Wavelet (DE-588)4215427-3 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Harmonische Analyse (DE-588)4023453-8 s DE-604 Wavelet (DE-588)4215427-3 s Graduate studies in mathematics 102 (DE-604)BV009739289 102 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017379162&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Pinsky, Mark A. 1940- Introduction to fourier analysis and wavelets Graduate studies in mathematics Fourier analysis Wavelets (Mathematics) Harmonische Analyse (DE-588)4023453-8 gnd Wavelet (DE-588)4215427-3 gnd |
| subject_GND | (DE-588)4023453-8 (DE-588)4215427-3 (DE-588)4151278-9 |
| title | Introduction to fourier analysis and wavelets |
| title_auth | Introduction to fourier analysis and wavelets |
| title_exact_search | Introduction to fourier analysis and wavelets |
| title_full | Introduction to fourier analysis and wavelets Mark A. Pinsky |
| title_fullStr | Introduction to fourier analysis and wavelets Mark A. Pinsky |
| title_full_unstemmed | Introduction to fourier analysis and wavelets Mark A. Pinsky |
| title_short | Introduction to fourier analysis and wavelets |
| title_sort | introduction to fourier analysis and wavelets |
| topic | Fourier analysis Wavelets (Mathematics) Harmonische Analyse (DE-588)4023453-8 gnd Wavelet (DE-588)4215427-3 gnd |
| topic_facet | Fourier analysis Wavelets (Mathematics) Harmonische Analyse Wavelet Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017379162&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009739289 |
| work_keys_str_mv | AT pinskymarka introductiontofourieranalysisandwavelets |