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  傅里叶分析及其应...
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• Contents
  Preface v
  1 Introduction 1
  1.1 The classical partial different requations 1
  1.2 Well-posed problems 3
  1.3 The one-dimensional wave equation 5
  1.4 Fourieris method 8
  2 Preparations 15
  2.1 Complex exponentials 15
  2.2 Complex-valued functions of a real variable 17
  2.3 Cesaro summation of series 20
  2.4 Positive summation kernels 22
  2.5 The Riemann-Lebesgue Iemma 25
  2.6 *Some simple distributions 27
  2.7 *Computing with 6 32
  3 Laplace and Z transforms 39
  3.i The Laplace transform 39
  3.2 0perations 42
  3.3 AppEications to differential equations 47
  3.4 Convolution 53
  3.5 *Laplace transforms of distributions 57
  3.6 The Z transform 60
  3.7 Applications in control theory 67
  Summary of Chapter 3 70
  4 Fourier series 73
  4.2 Dirichlet's and Fejer's kernels; uniqueness 80
  4.3 Differentiable functions 84
  4.4 Pointwise convergence 86
  4.5 Formulae for other periods 90
  4.6 Some worked examples 91
  4.7 The Gibbs phenomenon 93
  4.8 cFourier series for distributions 96
  Summary of Chapter 4 100
  5 L2 Theory 105
  5.1 Linear spaco over the complex numloers 105
  5.2 0rthogonal projections 114
  5.3 Fourier series for 119
  5.4 The Fourier system is complete 120
  5.5 Legendre polynomials 123
  5.6 0ther classical oflhogonal polynomials 127
  Summary of Chapter 5 130
  6 Separation of variables 137
  6.1 The solution of Fourier's problem 137
  6.2 Variations on Fourier's theme 139
  6.3 The Dirichlet problem in the unit disk 148
  6.4 Sturm-Liouville problems 153
  6.fi Some singulter Sturm-Liouville problems 159
  Summary ofChapter 6 160
  7 Fourier transforms 165
  7.2 Definition of the Fourier transform 166
  7.3 Properties 171
  7.4 The inversion theorem 168
  7.5 The convolution theorem 176
  7.6 Plancherel's formula 180
  7.7 Application 1 182
  7.8 Application 2 185
  7.9 Application 3: The sampling theorem 187
  7.10 *Connection with the Laplace transform 188
  7.11 ~Distributions and Fourier transforms 190
  Summary ofChapter 7 192
  8 Distributions 197
  8.1 History 197
  8.2 Fuzzy points test functions 200
  8.3 Distributions 203
  8.4 Properties 206
  8.5 Fourier transformation 213
  8.6 Convolution 218
  8.7 Periodic distributions and Fourier series 220
  8.8 Fttndamental solutions 221
  8.9 Back to the starting point 223
  Summary of Chapter 8 224
  9 Multi-dimensional Fourier analysis 227
  9.1 Rearranging series 227
  9.2 Double series 230
  9.3 Multi-dimensional Fourier series 233
  9.4 Multi-dimensional Fourier transforms 236
  Appendioes
  A The ubiquitous convolution 239
  B The discrete Fourier transform 243
  C Formulae 247
  C.1 Laplace transforms 247
  C.2 Z transforms 250
  C.3 Fourier series 251
  C.4 Fourier transforms 252
  C.5 Orthogonal polynomials 254
  D Answers to selected exercises 257
  ELiterature 265
  Index 267