在过去的二百年里，调和分析一直是最具影响力的数学思想之一，在理论意义上具有非凡的重要性，并在数学、科学和工程领域有广泛的应用。
本书旨在向高年级本科生和低年级研究生介绍调和分析的基础知识，从傅里叶（Fourier）对热方程的研究以及将函数展开为余弦和正弦之和（频率分析），到二进制调和分析以及将函数展开为哈尔（Haar）基函数之和（时间局部化）。作者传达了从傅里叶理论发展而来的思想所具有的非凡美感和适用性。尽管集中讨论了傅里叶和哈尔情形，但本书也涉及介于这两种不同函数展开方式之间的领域：时频分析（小波）。书中同时呈现了有限和连续的观点，引入了离散傅里叶和哈尔变换以及快速算法，如快速傅里叶变换（FFT）及其小波变换。
本书的方法结合了严格的证明、引人入胜的动机和丰富的应用。本书包含250多个练习题。每章结束时都会提供一些调和分析专题的研究思路，供学生独立完成。

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