This comprehensive volume develops all of the standard features of Fourier analysis - Fourier series, Fourier transform, Fourier sine and cosine transforms, and wavelets. The books approach emphasizes the role of the "selector" functions, and is not embedded in the usual engineering context, which makes the material more accessible to a wider audience. While there are several publications on the various individual topics, none combine or even include all of the above.

Inhalt

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.