调和分析:从傅里叶到小波

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