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Harmonic Analysis: From Fourier to Wavelets [Pehme köide]

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  • Formaat: Paperback / softback, 410 pages, kaal: 515 g
  • Sari: Student Mathematical Library
  • Ilmumisaeg: 30-Jul-2012
  • Kirjastus: American Mathematical Society
  • ISBN-10: 0821875663
  • ISBN-13: 9780821875667
Teised raamatud teemal:
Harmonic Analysis: From Fourier to WaveletsSuurem pilt
  • Formaat: Paperback / softback, 410 pages, kaal: 515 g
  • Sari: Student Mathematical Library
  • Ilmumisaeg: 30-Jul-2012
  • Kirjastus: American Mathematical Society
  • ISBN-10: 0821875663
  • ISBN-13: 9780821875667
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780821875667.html
Märksõnad:
In the last 200 years, harmonic analysis has been one of the most influential bodies of mathematical ideas, having been exceptionally significant both in its theoretical implications and in its enormous range of applicability throughout mathematics, science, and engineering. In this book, the authors convey the remarkable beauty and applicability of the ideas that have grown from Fourier theory. They present for an advanced undergraduate and beginning graduate student audience the basics of harmonic analysis, from Fourier's study of the heat equation, and the decomposition of functions into sums of cosines and sines (frequency analysis), to dyadic harmonic analysis, and the decomposition of functions into a Haar basis (time localization). While concentrating on the Fourier and Haar cases, the book touches on aspects of the world that lies between these two different ways of decomposing functions: time-frequency analysis (wavelets). Both finite and continuous perspectives are presented, allowing for the introduction of discrete Fourier and Haar transforms and fast algorithms, such as the Fast Fourier Transform (FFT) and its wavelet analogues. The approach combines rigorous proof, inviting motivation, and numerous applications. Over 250 exercises are included in the text. Each chapter ends with ideas for projects in harmonic analysis that students can work on independently. This book is published in cooperation with IAS/Park City Mathematics Institute.

Arvustused

The presentation of the material is very clear and illustrated by a number of enlightening figures. Many motivating remarks and discussions are provided. A number of proofs in the more elementary chapters are omitted, but precise pointers to the literature are given. Also numerous exercises are posed as well as some more involved 'projects' which motivate the reader to get active herself." - R. Steinbauer, Monatshefte für Mathematik

"This is a gentle introduction to Fourier analysis and wavelet theory that requires little background but still manages to explain some of the applications of Fourier and wavelet methods and touch on several current research topics. ... The authors have taken care to be accessible to undergraduate mathematicians. ... Compared to standard texts, this book is characterised by more personal and historical information, including footnotes. ... It comes with many projects for interested students, and lists a number of open-ended problems that suggest further developments and should engage interested students. ... In summary, this is a well-written and lively introduction to harmonic analysis that is accessible and stimulating for undergraduates and instructive and amusing for the more sophisticated reader. It could also be argued that the material herein should be part of the knowledge of most undergraduates in mathematics, given that the modern world relies more and more on data compression. It is therefore timely as well. It has certainly earned my enthusiastic recommendation." - Michael Cowling, Gazette of the Australian Mathematical Society

"A wonderful introduction to harmonic analysis and applications. The book is intended for advanced undergraduate and beginning graduate students and it is right on target. Pereyra and Ward present in a captivating style a substantial amount of classical Fourier analysis as well as techniques and ideas leading to current research. ... It is a great achievement to be able to present material at this level with only a minimal prerequisite of advanced calculus and linear algebra and a set of Useful Tools included in the appendix. I recommend this excellent book with enthusiasm and I encourage every student majoring in math to take a look." - Florin Catrina, MAA Reviews

"[ T]he panorama of harmonic analysis presented in the book includes very recent achievements like the connection of the dyadic shift operator with the Hilbert transform. This gives to an interested reader a good chance to see concrete examples of contemporary research problems in harmonic analysis. I highly recommend this book as a good source for undergraduate and graduate courses as well as for individual studies." - Krzysztof Stempak, Zentralblatt MATH

List of figures
xi
List of tables
xiii
IAS/Park City Mathematics Institute xv
Preface xvii
Suggestions for instructors xx
Acknowledgements xxii
Chapter 1 Fourier series: Some motivation
1(20)
§1.1 An example: Amanda calls her mother
2(5)
§1.2 The main questions
7(3)
§1.3 Fourier series and Fourier coefficients
10(5)
§1.4 History, and motivation from the physical world
15(4)
§1.5 Project: Other physical models
19(2)
Chapter 2 Interlude: Analysis concepts
21(34)
§2.1 Nested classes of functions on bounded intervals
22(16)
§2.2 Modes of convergence
38(7)
§2.3 Interchanging limit operations
45(4)
§2.4 Density
49(3)
§2.5 Project: Monsters, Take I
52(3)
Chapter 3 Pointwise convergence of Fourier series
55(22)
§3.1 Pointwise convergence: Why do we care?
55(4)
§3.2 Smoothness vs. convergence
59(9)
§3.3 A suite of convergence theorems
68(5)
§3.4 Project: The Gibbs phenomenon
73(1)
§3.5 Project: Monsters, Take II
74(3)
Chapter 4 Summability methods
77(30)
§4.1 Partial Fourier sums and the Dirichlet kernel
78(4)
§4.2 Convolution
82(8)
§4.3 Good kernels, or approximations of the identity
90(6)
§4.4 Fejer kernels and Cesaro means
96(3)
§4.5 Poisson kernels and Abel means
99(2)
§4.6 Excursion into LP(T)
101(1)
§4.7 Project: Weyl's Equidistribution Theorem
102(2)
§4.8 Project: Averaging and summability methods
104(3)
Chapter 5 Mean-square convergence of Fourier series
107(20)
§5.1 Basic Fourier theorems in L2(T)
108(2)
§5.2 Geometry of the Hilbert space L2(T)
110(5)
§5.3 Completeness of the trigonometric system
115(7)
§5.4 Equivalent conditions for completeness
122(4)
§5.5 Project: The isoperimetric problem
126(1)
Chapter 6 A tour of discrete Fourier and Haar analysis
127(34)
§6.1 Fourier series vs. discrete Fourier basis
128(6)
§6.2 Short digression on dual bases in CN
134(3)
§6.3 The Discrete Fourier Transform and its inverse
137(1)
§6.4 The Fast Fourier Transform (FFT)
138(8)
§6.5 The discrete Haar basis
146(5)
§6.6 The Discrete Haar Transform
151(1)
§6.7 The Fast Haar Transform
152(5)
§6.8 Project: Two discrete Hilbert transforms
157(2)
§6.9 Project: Fourier analysis on finite groups
159(2)
Chapter 7 The Fourier transform in paradise
161(28)
§7.1 From Fourier series to Fourier integrals
162(2)
§7.2 The Schwartz class
164(3)
§7.3 The time-frequency dictionary for S(R)
167(5)
§7.4 The Schwartz class and the Fourier transform
172(3)
§7.5 Convolution and approximations of the identity
175(4)
§7.6 The Fourier Inversion Formula and Plancherel
179(5)
§7.7 LP norms on S(R)
184(3)
§7.8 Project: A bowl of kernels
187(2)
Chapter 8 Beyond paradise
189(32)
§8.1 Continuous functions of moderate decrease
190(3)
§8.2 Tempered distributions
193(4)
§8.3 The time-frequency dictionary for S'(R)
197(5)
§8.4 The delta distribution
202(3)
§8.5 Three applications of the Fourier transform
205(8)
§8.6 LP(R) as distributions
213(4)
§8.7 Project: Principal value distribution 1/x
217(1)
§8.8 Project: Uncertainty and primes
218(3)
Chapter 9 From Fourier to wavelets, emphasizing Haar
221(40)
§9.1 Strang's symphony analogy
222(2)
§9.2 The windowed Fourier and Gabor bases
224(6)
§9.3 The wavelet transform
230(6)
§9.4 Haar analysis
236(14)
§9.5 Haar vs. Fourier
250(7)
§9.6 Project: Local cosine and sine bases
257(1)
§9.7 Project: Devil's advocate
257(1)
§9.8 Project: Khinchine's Inequality
258(3)
Chapter 10 Zooming properties of wavelets
261(42)
§10.1 Multiresolution analyses (MRAs)
262(9)
§10.2 Two applications of wavelets, one case study
271(7)
§10.3 From MRA to wavelets: Mallat's Theorem
278(10)
§10.4 How to find suitable MRAs
288(10)
§10.5 Projects: Twin dragon; infinite mask
298(1)
§10.6 Project: Linear and nonlinear approximations
299(4)
Chapter 11 Calculating with wavelets
303(26)
§11.1 The Haar multiresolution analysis
303(5)
§11.2 The cascade algorithm
308(6)
§11.3 Filter banks, Fast Wavelet Transform
314(9)
§11.4 A wavelet library
323(5)
§11.5 Project: Wavelets in action
328(1)
Chapter 12 The Hilbert transform
329(42)
§12.1 In the frequency domain: A Fourier multiplier
330(3)
§12.2 In the time domain: A singular integral
333(3)
§12.3 In the Haar domain: An average of Haar shifts
336(5)
§12.4 Boundedness on LP of the Hilbert transform
341(5)
§12.5 Weak boundedness on L1(R)
346(6)
§12.6 Interpolation and a festival of inequalities
352(6)
§12.7 Some history to conclude our journey
358(7)
§12.8 Project: Edge detection and spectroscopy
365(1)
§12.9 Projects: Harmonic analysis for researchers
366(5)
Appendix. Useful tools
371(20)
§A.1 Vector spaces, norms, inner products
371(6)
§A.2 Banach spaces and Hilbert spaces
377(5)
§A.3 LP(R), density, interchanging limits on R
382(9)
Bibliography 391(10)
Name index 401(2)
Subject index 403
María Cristina Pereyra, The University of New Mexico, Albuquerque, NM, USA

Lesley A. Ward, University of South Australia, Mawson Lakes Campus, Adelaide, Australia