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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