1 What are Distributions? 1
1.1 Generalized functions and test functions 1
Contents 5
Preface 5
1.2 Examples of distributions 5
1.3 What good are distributions? 8
1.4 Problems 10
2 The Calculus of Distributions 12
2.1 Functions as distributions 12
2.2 Operations on distributions 14
2.3 Adioint identities 18
2.4 Consistency of derivatives 20
2.5 Distributional solutions of differential equations 22
2.6 Problems 25
3.1 From Fourier series to Fourier integrals 28
3 Fourier Transforms 28
3.2 The Schwartz class S 31
3.3 Properties of the Fourier transform on S 32
3.4 The Fourier inversion formula on S 38
3.5 The Fourier transform of a Gaussian 41
3.6 Problems 43
4 Fourier Transforms of Tempered Distributions 46
4.1 The definitions 46
4.2 Examples 49
4.3 Convolutions with tempered distributions 55
4.4 Problems 57
5 Solving Partial Differential Equations 60
5.1 The Laplace equation 60
5.2 The heat equation 64
5.3 The wave equation 67
5.4 Schr？dinger's equation and quantum mechanics 72
5.5 Problems 73
6 The Structure of Distributions 78
6.1 The support of a distribution 78
6.2 Structure theorems 82
6.3 Distributions with point support 85
6.4 Positive distributions 88
6.5 Continuity of distribution 91
6.6 Approximation by test functions 98
6.7 Local theory of distributions 101
6.8 Distributions on spheres 103
6.9 Problems 108
7 Fourier Analysis 113
7.1 The Riemann-Lebesgue lemma 113
7.2 Paley-Wiener theorems 119
7.3 The Poisson summation formula 125
7.4 Probability measures and positive definite functions 130
7.5 The Heisenberg uncertainty principle 134
7.6 Hermite functions 139
7.7 Radial Fourier transforms and Bessel functions 143
7.8 Haar functions and wavelets 149
7.9 Problems 157
8 Sobolev Theory and Microlocal Analysis 162
8.1 Sobolev inequalities 162
8.2 Sobolev spaces 172
8.3 Elliptic partial differential equations(constant coefficients) 176
8.4 Pseudodifferential operators 185
8.5 Hyperbolic operators 191
8.6 The wave front set 200
8.7 Microlocal analysis of singularities 209
8.8 Problems 214
Suggestions for Further Reading 219
Index 221