Author | Bachman, George. author |
---|---|
Title | Fourier and Wavelet Analysis [electronic resource] / by George Bachman, Lawrence Narici, Edward Beckenstein |
Imprint | New York, NY : Springer New York : Imprint: Springer, 2000 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-0505-0 |
Descript | IX, 507 p. online resource |
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 โ Divergent Fourier Series -- 4.10 Termwise Integration -- 4.11 Trigonometric vs. Fourier Series -- 4.12 Termwise Differentiation -- 4.13 Didoโs Dilemma -- 4.14 Other Kinds of Summability -- 4.15 Fejer Theory -- 4.16 The Smoothing Effect of (C, 1) Summation -- 4.17 Weierstrassโs Approximation Theorem -- 4.18 Lebesgueโs 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 Lebesgueโs 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 Parsevalโs Identities -- 5.18 The L2Theory -- 5.19 The Plancherel Theorem -- 5.20 Point wise Inversion and Summability -- 5.21 โ 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 โ 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