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This book is intended as an introduction to classical Fourier analysis, Fourier series, the Fourier transform, and Fourier sine and cosine transforms. The topics are developed slowly for the reader who has never seen them before, with a preference for clarity of exposition over elegance in stating and proving results. More recent developments, such as the discrete and fast Fourier transforms and wavelets, are covered in the last two chapters, where the approach is elementary, intended for the novice reader. The first three, short, chapters present requisite background material, and these could be read as a short course in functional analysis. The text includes many historical notes to place the material in a cultural and mathematical context; from the fact that Jean Baptiste Joseph Fourier was the nineteenth, but not the last, child in his family to the impact that Fourier series have had on the evolution of the concept of the integral.

Fourier and Wavelet Analysis

1 Metrie and Normed Spaces.- 1.1 Metrie Spaces.- 1.2 Normed Spaces.- 1.3 Inner Product Spaces.- 1.4 Orthogonality.- 1.5 Linear Isometry.- 1.6 Holder and Minkowski Inequalities; Lpand lpSpaces..- 2 Analysis.- 2.1 Balls.- 2.2 Convergence and Continuity.- 2.3 Bounded Sets.- 2.4 Closure and Closed Sets.- 2.5 Open Sets.- 2.6 Completeness.- 2.7 Uniform Continuity.- 2.8 Compactness.- 2.9 Equivalent Norms.- 2.10 Direct Sums.- 3 Bases.- 3.1 Best Approximation.- 3.2 Orthogonal Complements and the Projection Theorem.- 3.3 Orthonormal Sequences.- 3.4 Orthonormal Bases.- 3.5 The Haar Basis.- 3.6 Unconditional Convergence.- 3.7 Orthogonal Direct Sums.- 3.8 Continuous Linear Maps.- 3.9 Dual Spaces.- 3.10 Adjoints.- 4 Fourier Series.- 4.1 Warmup.- 4.2 Fourier Sine Series and Cosine Series.- 4.3 Smoothness.- 4.4 The Riemann-Lebesgue Lemma.- 4.5 The Dirichlet and Fourier Kernels.- 4.6 Point wise Convergence of Fourier Series.- 4.7 Uniform Convergence.- 4.8 The Gibbs Phenomenon.- 4.9 - Divergent Fourier Series.- 4.10 Termwise Integration.- 4.11 Trigonometric vs. Fourier Series.- 4.12 Termwise Differentiation.- 4.13 Dido's Dilemma.- 4.14 Other Kinds of Summability.- 4.15 Fejer Theory.- 4.16 The Smoothing Effect of (C, 1) Summation.- 4.17 Weierstrass's Approximation Theorem.- 4.18 Lebesgue's Pointwise Convergence Theorem.- 4.19 Higher Dimensions.- 4.20 Convergence of Multiple Series.- 5 The Fourier Transform.- 5.1 The Finite Fourier Transform.- 5.2 Convolution on T.- 5.3 The Exponential Form of Lebesgue's Theorem.- 5.4 Motivation and Definition.- 5.5 Basics/Examplesv.- 5.6 The Fourier Transform and Residues.- 5.7 The Fourier Map.- 5.8 Convolution on R.- 5.9 Inversion, Exponential Form.- 5.10 Inversion, Trigonometric Form.- 5.11 (C, 1) Summability for Integrals.- 5.12 The Fejer-Lebesgue Inversion Theorem.- 5.13 Convergence Assistance.- 5.14 Approximate Identity.- 5.15 Transforms of Derivatives and Integrals.- 5.16 Fourier Sine and Cosine Transforms.- 5.17 Parseval's Identities.- 5.18 The L2Theory.- 5.19 The Plancherel Theorem.- 5.20 Point wise Inversion and Summability.- 5.21 - Sampling Theorem.- 5.22 The Mellin Transform.- 5.23 Variations.- 6 The Discrete and Fast Fourier Transforms.- 6.1 The Discrete Fourier Transform.- 6.2 The Inversion Theorem for the DFT.- 6.3 Cyclic Convolution.- 6.4 Fast Fourier Transform for N=2k.- 6.5 The Fast Fourier Transform for N=RC.- 7 Wavelets.- 7.1 Orthonormal Basis from One Function.- 7.2 Multiresolution Analysis.- 7.3 Mother Wavelets Yield Wavelet Bases.- 7.4 From MRA to Mother Wavelet.- 7.5 Construction of - Scaling Function with Compact Support.- 7.6 Shannon Wavelets.- 7.7 Riesz Bases and MRAs.- 7.8 Franklin Wavelets.- 7.9 Frames.- 7.10 Splines.- 7.11 The Continuous Wavelet Transform.