Fourier Analysis and Its Applications

Contents......Page 10
Preface......Page 8
1.1 The classical partial differential equations......Page 14
1.2 Well-posed problems......Page 16
1.3 The one-dimensional wave equation......Page 18
1.4 Fourier’s method......Page 22
2.1 Complex exponentials......Page 28
2.2 Complex-valued functions of a real variable......Page 30
2.3 Cesàro summation of series......Page 33
2.4 Positive summation kernels......Page 35
2.5 The Riemann–Lebesgue lemma......Page 38
2.6 *Some simple distributions......Page 40
2.7 *Computing with δ......Page 45
3.1 The Laplace transform......Page 52
3.2 Operations......Page 55
3.3 Applications to differential equations......Page 60
3.4 Convolution......Page 66
3.5 *Laplace transforms of distributions......Page 70
3.6 The Z transform......Page 73
3.7 Applications in control theory......Page 80
Summary of Chapter 3......Page 83
4.1 Definitions......Page 86
4.2 Dirichlet’s and Fejér’s kernels; uniqueness......Page 93
4.3 Differentiable functions......Page 97
4.4 Pointwise convergence......Page 99
4.5 Formulae for other periods......Page 103
4.6 Some worked examples......Page 104
4.7 The Gibbs phenomenon......Page 106
4.8 *Fourier series for distributions......Page 109
Summary of Chapter 4......Page 113
5.1 Linear spaces over the complex numbers......Page 118
5.2 Orthogonal projections......Page 123
5.3 Some examples......Page 127
5.4 The Fourier system is complete......Page 132
5.5 Legendre polynomials......Page 136
5.6 Other classical orthogonal polynomials......Page 140
Summary of Chapter 5......Page 143
6.1 The solution of Fourier’s problem......Page 150
6.2 Variations on Fourier’s theme......Page 152
6.3 The Dirichlet problem in the unit disk......Page 161
6.4 Sturm–Liouville problems......Page 166
6.5 Some singular Sturm–Liouville problems......Page 172
Summary of Chapter 6......Page 173
7.1 Introduction......Page 178
7.2 Definition of the Fourier transform......Page 179
7.3 Properties......Page 181
7.4 The inversion theorem......Page 184
7.5 The convolution theorem......Page 189
7.6 Plancherel’s formula......Page 193
7.7 Application 1......Page 195
7.8 Application 2......Page 198
7.9 Application 3: The sampling theorem......Page 200
7.10 *Connection with the Laplace transform......Page 201
7.11 *Distributions and Fourier transforms......Page 203
Summary of Chapter 7......Page 205
8.1 History......Page 210
8.2 Fuzzy points – test functions......Page 213
8.3 Distributions......Page 216
8.4 Properties......Page 219
8.5 Fourier transformation......Page 226
8.6 Convolution......Page 231
8.7 Periodic distributions and Fourier series......Page 233
8.8 Fundamental solutions......Page 234
8.9 Back to the starting point......Page 236
Summary of Chapter 8......Page 237
9.1 Rearranging series......Page 240
9.2 Double series......Page 243
9.3 Multi-dimensional Fourier series......Page 246
9.4 Multi-dimensional Fourier transforms......Page 249
A: The ubiquitous convolution......Page 252
B: The discrete Fourier transform......Page 256
C.1 Laplace transforms......Page 260
C.2 Z transforms......Page 263
C.3 Fourier series......Page 264
C.4 Fourier transforms......Page 265
C.5 Orthogonal polynomials......Page 267
D: Answers to selected exercises......Page 270
E: Literature......Page 278
E......Page 280
P......Page 281
Z......Page 282