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| Preface | |
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| What are Distributions? | |
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| Generalized functions and test functions | |
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| Examples of distributions | |
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| What good are distributions? | |
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| Problems | |
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| The Calculus of Distributions | |
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| Functions as distributions | |
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| Operations on distributions | |
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| Adjoint identities | |
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| Consistency of derivatives | |
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| Distributional solutions of differential equations | |
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| Problems | |
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| Fourier Transforms | |
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| From Fourier series to Fourier integrals | |
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| The Schwartz class S | |
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| Properties of the Fourier transform on S | |
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| The Fourier inversion formula on S | |
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| The Fourier transform of a Gaussian | |
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| Problems | |
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| Fourier Transforms of Tempered Distributions | |
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| The definitions | |
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| Examples | |
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| Convolutions with tempered distributions | |
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| Problems | |
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| Solving Partial Differential Equations | |
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| The Laplace equation | |
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| The heat equation | |
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| The wave equation | |
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| Schrodinger's equation and quantum mechanics | |
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| Problems | |
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| The Structure of Distributions | |
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| The support of a distribution | |
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| Structure theorems | |
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| Distributions with point support | |
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| Positive distributions | |
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| Continuity of distribution | |
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| Approximation by test functions | |
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| Local theory of distributions | |
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| Distributions on spheres | |
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| Problems | |
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| Fourier Analysis | |
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| The Riemann-Lebesgue lemma | |
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| Paley-Wiener theorems | |
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| The Poisson summation formula | |
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| Probability measures and positive definite functions | |
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| The Heisenberg uncertainty principle | |
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| Hermite functions | |
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| Radial Fourier transforms and Bessel functions | |
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| Haar functions and wavelets | |
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| Problems | |
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| Sobolev Theory and Microlocal Analysis | |
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| Sobolev inequalities | |
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| Sobolev spaces | |
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| Elliptic partial differential equations (constant coefficients) | |
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| Pseudodifferential operators | |
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| Hyperbolic operators | |
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| The wave front set | |
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| Microlocal analysis of singularities | |
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| Problems | |
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| Suggestions for Further Reading | |
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| Index | |