調和分析-從傅里葉到小波 (影印版) 9787040612011 (美)瑪利亞.克里斯蒂娜.佩雷拉 (澳)萊拉.A.

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書名：調和分析-從傅里葉到小波 (影印版)
ISBN：9787040612011
出版社：高等教育
著編譯者：(美)瑪利亞.克里斯蒂娜.佩雷拉 (澳)萊拉.A. 沃德著
叢書名：美國數學會經典影印系列
頁數：448
所在地：中國大陸 *此為代購商品
書號：1628662
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【台灣高等教育出版社簡體書】調和分析-從傅里葉到小波 (影印版) 787040612011 (美)瑪利亞.克里斯蒂娜.佩雷拉 (澳)萊拉.A. 沃德著

內容簡介

在過去的二百年裡，調和分析一直是最具影響力的數學思想之一，在理論意義上具有非凡的重要性，並在數學、科學和工程領域有廣泛的應用。本書旨在向高年級本科生和低年級研究生介紹調和分析的基礎知識，從傅里葉（Fourier）對熱方程的研究以及將函數展開為餘弦和正弦之和（頻率分析），到二進位調和分析以及將函數展開為哈爾（Haar）基函數之和（時間局部化）。作者傳達了從傅里葉理論發展而來的思想所具有的非凡美感和適用性。儘管集中討論了傅里葉和哈爾情形，但本書也涉及介於這兩種不同函數展開方式之間的領域：時頻分析（小波）。書中同時呈現了有限和連續的觀點，引入了離散傅里葉和哈爾變換以及快速演算法，如快速傅里葉變換（FFT）及其小波變換。本書的方法結合了嚴格的證明、引人入勝的動機和豐富的應用。本書包含250多個練習題。每章結束時都會提供一些調和分析專題的研究思路，供學生獨立完成。

List of figures
List of tables
IAS/Park City Mathematics Institute
Preface
Suggestions for instructors
Acknowledgements
Chapter 1 Fourier series: Some motivation
§1 1 An example: Amanda calls her mother
§1 2 The main questions
§1 3 Fourier series and Fourier coefficients
§1 4 History， and motivation from the physical world
§1 5 Project: Other physical models
Chapter 2 Interlude: Analysis concepts
§2 1 Nested classes of functions on bounded intervals
§2 2 Modes of convergence
§2 3 Interchanging limit operations
§2 4 Density
§2 5 Project: Monsters， Take I
Chapter 3 Pointwise convergence of Fourier series
§3 1 Pointwise convergence: Why do we care?
§3 2 Smoothness vs convergence
§3 3 A suite of convergence theorems
§3 4 Project: The Gibbs phenomenon
§3 5 Project: Monsters， Take II
Chapter 4 Summability methods
§4 1 Partial Fourier sums and the Dirichlet kernel
§4 2 Convolution
§4 3 Good kernels， or approximations of the identity
§4 4 Fejer kernels and CesAro means
§4 5 Poisson kernels and Abel means
§4 6 Excursion into LP(T)
§4 7 Project: Weyl's Equidistribution Theorem
§4 8 Project: Averaging and summability methods
Chapter 5 Mean-square convergence of Fourier series
§5 1 Basic Fourier theorems in L2(T)
§5 2 Geometry of the Hilbert space L2(T)
§5 3 Completeness of the trigonometric system
§5 4 Equivalent conditions for completeness
§5 5 Project: The isoperimetric problem
Chapter 6 A tour of discrete Fourier and Haar analysis
§6 1 Fourier series vs discrete Fourier basis
§6 2 Short digression on dual bases in CN
§6 3 The Discrete Fourier Transform and its inverse
§6 4 The Fast Fourier Transform (FFT)
§6 5 The discrete Haar basis
§6 6 The Discrete Haar Transform
§6 7 The Fast Haar Transform
§6 8 Project: Two discrete Hilbert transforms
§6 9 Project: Fourier analysis on finite groups
Chapter 7 The Fourier transform in paradise
§7 1 From Fourier series to Fourier integrals
§7 2 The Schwartz class
§7 3 The time-frequency dictionary for S(R)
§7 4 The Schwartz class and the Fourier transform
§7 5 Convolution and approximations of the identity
§7 6 The Fourier Inversion Formula and Plancherel
§7 7 Lp norms on S(R)
§7 8 Project: A bowl of kernels
Chapter 8 Beyond paradise
§8 1 Continuous functions of moderate decrease
§8 2 Tempered distributions
§8 3 The time-frequency dictionary for S'(R)
§8 4 The delta distribution
§8 5 Three applications of the Fourier transform
§8 6 Lp(R) as distributions
§8 7 Project: Principal value distribution 1/x
§8 8 Project: Uncertainty and primes
Chapter 9 From Fourier to wavelets， emphasizing Haar
§9 1 Strang's symphony analogy
§9 2 The windowed Fourier and Gabor bases
§9 3 The wavelet transform
§9 4 Haar analysis
§9 5 Haar vs Fourier
§9 6 Project: Local cosine and sine bases
§9 7 Project: Devil's advocate
§9 8 Project: Khinchine's Inequality
Chapter 10 Zooming properties of wavelets
§10 1 Multiresolution analyses (MRAs)
§10 2 Two applications of wavelets， one case study
§10 3 From MRA to wavelets: Mallat's Theorem
§10 4 How to find suitable MRAs
§10 5 Projects: Twin dragon; infinite mask
§10 6 Project: Linear and nonlinear approximations
Chapter 11 Calculating with wavelets
§11 1 The Haar multiresolution analysis
§11 2 The cascade algorithm
§11 3 Filter banks， Fast Wavelet Transform
§11 4 A wavelet library
§11 5 Project: Wavelets in action
Chapter 12 The Hilbert transform
§12 1 In the frequency domain: A Fourier multiplier
§12 2 In the time domain: A singular integral
§12 3 In the Haar domain: An average of Haar shifts
§12 4 Boundedness on Lp of the Hilbert transform
§12 5 Weak boundedness on L1(R)
§12 6 Interpolation and a festival of inequalities
§12 7 Some history to
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