Chapter 1．The Genesis of Fourier Analysis 1
1 The vibrating string 2
1.1 Derivation of the wave equation 6
1.2 Solution to the wave equation 8
1.3 Example: the plucked string 17
2 The heat equation 18
2.1 Derivation of the heat equation 18
2.2 Steady-state heat equation in the disc 20
3 Exercises 23
4 Problem 28
Chapter 2．Basic Properties of Fourier Series 29
1 Examples and formulation of the problem 30
1.1 Main definitions and some examples 34
2 Uniqueness of Fourier series 39
3 Convolutions 44
4 Good kernels 48
5 Cesàro and Abel summability: applications to Fourier series 51
5.1 Cesàro means and summation 51
5.2 Fejér's theorem 52
5.3 Abel means and summation 54
5.4 The Poisson kernel and Dirichlet's problem in the unit disc 55
6 Exercises 58
7 Problems 65
Chapter 3．Convergence of Fourier Series 69
1 Mean-square convergence of Fourier series 70
1.1 Vector spaces and inner products 70
1.2 Proof of mean-square convergence 76
2 Return to pointwise convergence 81
2.1 A local result 81
2.2 A continuous function with diverging Fourier series 83
3 Exercises 87
4 Problems 95
Chapter 4．Some Applications of Fourier Series 100
1 The isoperimetric inequality 101
2 Weyl's equidistribution theorem 105
3 A continuous but nowhere differentiable function 113
4 The heat equation on the circle 118
5 Exercises 120
6 Problems 125
Chapter 5．The Fourier Transform on ？ 129
1 Elementary theory of the Fourier transform 131
1.1 Integration of functions on the real line 131
1.2 Definition of the Fourier transform 134
1.3 The Schwartz space 134
1.4 The Fourier transform on S 136
1.5 The Fourier inversion 140
1.6 The Plancherel formula 142
1.7 Extension to functions of moderate decrease 144
1.8 The Weierstrass approximation theorem 144
2 Applications to some partial differential equations 145
2.1 The time-dependent heat equation on the real line 145
2.2 The steady-state heat equation in the upper half-plane 149
3 The Poisson summation formula 153
3.1 Theta and zeta functions 155
3.2 Heat kernels 156
3.3 Poisson kernels 157
4 The Heisenberg uncertainty principle 158
5 Exercises 161
6 Problems 169
Chapter 6．The Fourier Transform on ？d 175
1 Preliminaries 176
1.1 Symmetries 176
1.2 Integration on ？d 178
2 Elementary theory of the Fourier transform 180
3 The wave equation in ？d × ？ 184
3.1 Solution in terms of Fourier transforms 184
3.2 The wave equation in ？3 × ？ 189
3.3 The wave equation in ？2 × ？: descent 194
4 Radial symmetry and Bessel functions 196
5 The Radon transform and some of its applications 198
5.1 The X-ray transform in ？2 199
5.2 The Radon transform in ？3 201
5.3 A note about plane waves 207
6 Exercises 207
7 Problems 212
Chapter 7．Finite Fourier Analysis 218
1 Fourier analysis on ？(N) 219
1.1 The group ？(N) 219
1.2 Fourier inversion theorem and Plancherel identity on ？(N) 221
1.3 The fast Fourier transform 224
2 Fourier analysis on finite abelian groups 226
2.1 Abelian groups 226
2.2 Characters 230
2.3 The orthogonality relations 232
2.4 Characters as a total family 233
2.5 Fourier inversion and Plancherel formula 235
3 Exercises 236
4 Problems 239
Chapter 8．Dirichlet's Theorem 241
1 A little elementary number theory 241
1.1 The fundamental theorem of arithmetic 241
1.2 The infinitude of primes 244
2 Dirichlet's theorem 252
2.1 Fourier analysis, Dirichlet characters, and reduc-tion of the theorem 254
2.2 Dirichlet L-functions 255
3 Proof of the theorem 258
3.1 Logarithms 258
3.2 L-functions 261
3.3 Non-vanishing of the L-function 265
4 Exercises 275
5 Problems 279
Appendix: Integration 281
1 Definition of the Riemann integral 281
1.1 Basic properties 282
1.2 Sets of measure zero and discontinuities of inte-grable functions 286
2 Multiple integrals 289
2.1 The Riemann integral in ？d 289
2.2 Repeated integrals 291
2.3 The change of variables formula 292
2.4 Spherical coordinates 293
3 Improper integrals. Integration over ？d 294
3.1 Integration of functions of moderate decrease 294
3.2 Repeated integrals 295
3.3 Spherical coordinates 297
Notes and References 299
Bibliography 301
Symbol Glossary 305