Preface 1 Introduction 1.1 The classical partial differential equations 1.2 Wall-posed problems 1.3 The one-dimensional wave equation 1.4 Fourier's method 2 Preparations 2.1 Complex exponentials 2.2 Complex-valued functions of a real variable 2.3 Cesàro summation of series 2.4 Positive summation kernels 2.5 The Riemann-Lebesgue lemma 2.6 *Some simple distributions 2.7 *Computing with δ 3 Laplace and Z transforms 3 Laplace and Z transforms 3.2 Operations 3.3 Applications to differential equations 3.4 Convolution 3.5 *Laplace transforms of distributions 3.6 The Z transform 3.7 Applications in control theory Summary of Chapter 3 4 Fourier series 4.1 Definitions 4.2 Dirichlet's and Fejér's kernels;uniqueness 4.3 Differentiable functions 4.4 Pointwise convergence 4.5 Formulae for other periods 4.6 Some worked examples 4.7 The Gibbs phenomenon 4.8 *Fourier series for distributions Smnmary of Chapter 4 5 L^2 Theory 5.1 Linear spaces over the complex numbers 5.2 Orthogonal projections 5.3 Some examples 5.4 The Fourier system is complete 5.5 Legendre polynomials 5.6 Other classical orthogonal polynomials Summary of Chapter 5 6 Separation of variables 6.1 The solution of Fourier's problem 6.2 Variations on Fourier's theme 6.3 The Dirichlet problem in the unit disk 6.4 Sturm-Liouvi]le problems 6.5 Some singular Sturm-Liouville problems Summary of Chapter 6 7 Fourier transforms 7.1 Lntreduction 7.2 Definition of the Fourier transform 7.3 Properties ?,4 The inversion theorem ?.5 The convolution theorem 7.6 plancherel's formula 7.7 Application l 7.8 Application 2 7.9 Application 3:The sampling theorem 7.10 *Connection with the Laplace transform 7.11 *Distributions and Fourier trausforms Summary of Chapter 7 8 Distributions 8.1 History 8.2 Fuzzy points-test functions 8.3 Distributions 8.4 Properties 8.5 Fourier transformation 8.6 Convolution 8.7 Periodic distributions and Fourier series 8.8 Fundamental solutions 8.9 Back to the starting point Summary of Chapter 8 9 Multi-dimensional Fourier analysis 9.1 Rearranging series 9.2 Double series 9.3 Multi-dimensional Fourier series 9.4 Multi-dimensional Fourier transforms Appendices A The ubiquitous convolution B The discrete Fourier transform C Formulae C.1 Lapisce transforms C.2 Z transforms C.3 Fourier series C.4 Fourier transforms C.5 Orthogonal polynomials D Answers to selected exercises E Literature Index
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