Fourier Series, Fourier Transform and Their Applications to Mathematical Physics

Preface......Page 6
Contents......Page 9
Part I Fourier Series and the Discrete Fourier Transform......Page 12
1 Introduction......Page 13
2 Formulation of Fourier Series......Page 21
3 Fourier Coefficients and Their Properties......Page 26
4 Convolution and Parseval\'s Equality......Page 32
5 Fejér Means of Fourier Series. Uniqueness  of the Fourier Series.......Page 35
6 The Riemann--Lebesgue Lemma......Page 40
7 The Fourier Series of a Square-Integrable Function. The Riesz--Fischer Theorem.......Page 43
8 Besov and Hölder Spaces......Page 51
9 Absolute Convergence. Bernstein and Peetre Theorems.......Page 58
10 Dirichlet Kernel. Pointwise and Uniform Convergence.......Page 64
11 Formulation of the Discrete Fourier Transform and Its Properties.......Page 81
12 Connection Between the Discrete Fourier Transform and the Fourier Transform.......Page 89
13 Some Applications of the Discrete Fourier Transform.......Page 97
 14.1 The One-Dimensional Heat Equation......Page 103
 14.2 The One-Dimensional Wave Equation......Page 117
 14.3 The Laplace Equation in a Rectangle and in a Disk......Page 125
Part II Fourier Transform and Distributions......Page 133
15 Introduction......Page 134
16 The Fourier Transform in Schwartz Space......Page 136
17 The Fourier Transform in Lp(mathbbRn), 1leqpleq2......Page 145
18 Tempered Distributions......Page 155
19 Convolutions in S and S\'......Page 168
20 Sobolev Spaces......Page 175
 20.1 Sobolev Spaces on Bounded Domains......Page 188
21 Homogeneous Distributions......Page 193
22 Fundamental Solution of the Helmholtz Operator......Page 206
23 Estimates for the Laplacian and Hamiltonian......Page 216
Part III Operator Theory and Integral Equations......Page 244
24 Introduction......Page 245
25 Inner Product Spaces and Hilbert Spaces......Page 247
26 Symmetric Operators in Hilbert Spaces......Page 258
27 John von Neumann\'s Spectral Theorem......Page 276
28 Spectra of Self-Adjoint Operators......Page 291
29 Quadratic Forms. Friedrichs Extension.......Page 308
30 Elliptic Differential Operators......Page 313
31 Spectral Functions......Page 325
32 The Schrödinger Operator......Page 329
33 The Magnetic Schrödinger Operator......Page 342
34 Integral Operators with Weak Singularities. Integral Equations of the First and Second Kinds.......Page 352
35 Volterra and Singular Integral Equations......Page 364
36 Approximate Methods......Page 372
Part IV Partial Differential Equations......Page 383
37 Introduction......Page 384
38 Local Existence Theory......Page 395
39 The Laplace Operator......Page 410
40 The Dirichlet and Neumann Problems......Page 425
41 Layer Potentials......Page 438
42 Elliptic Boundary Value Problems......Page 457
43 The Direct Scattering Problem for the Helmholtz Equation......Page 470
44 Some Inverse Scattering Problems for the Schrödinger Operator......Page 478
45 The Heat Operator......Page 492
46 The Wave Operator......Page 502
Appendix  References......Page 514
Index......Page 516