This book provides a concise exposition of the basic ideas of the theory of distribution and Fourier transforms and its application to partial differential equations. The author clearly presents the ideas, precise statements of theorems, and explanations of ideas behind the proofs. Methods in which techniques are used in applications are illustrated, and many problems are included. The book also introduces several significant recent topics, including pseudo-differential operators, wave front sets, wavelets, and quasicrystals. Background mathematical prerequisites have been kept to a minimum, with only a knowledge of multidimensional calculus and basic complex variables needed to fully understand the concepts in the book. in applied analysis and mathematical physics.
Author(s): Robert Strichartz
Series: Studies in Advanced Mathematics
Edition: 1
Publisher: CRC-Press
Year: 1994
Language: English
Pages: 222
Contents......Page all_9140_to_00222.cpc0003.djvu
Preface......Page all_9140_to_00222.cpc0005.djvu
1.1 Generalized functions and test functions......Page all_9140_to_00222.cpc0009.djvu
1.2 Examples of distributions......Page all_9140_to_00222.cpc0013.djvu
1.3 What good are distributions?......Page all_9140_to_00222.cpc0015.djvu
1.4 Problems......Page all_9140_to_00222.cpc0017.djvu
2.1 Functions as distributions......Page all_9140_to_00222.cpc0019.djvu
2.2 Operations on distributions......Page all_9140_to_00222.cpc0020.djvu
2.3 Adjoint identities......Page all_9140_to_00222.cpc0025.djvu
2.4 Consistency of derivatives......Page all_9140_to_00222.cpc0027.djvu
2.5 Distributional solutions of differential equations......Page all_9140_to_00222.cpc0029.djvu
2.6 Problems......Page all_9140_to_00222.cpc0031.djvu
3.1 From Fourier series to Fourier integrals......Page all_9140_to_00222.cpc0034.djvu
3.2 The Schwartz class S......Page all_9140_to_00222.cpc0037.djvu
3.3 Properties of the Fourier transform on S......Page all_9140_to_00222.cpc0038.djvu
3.4 The Fourier inversion formula on S......Page all_9140_to_00222.cpc0043.djvu
3.5 The Fourier transform of a Gaussian......Page all_9140_to_00222.cpc0046.djvu
3.6 Problems......Page all_9140_to_00222.cpc0049.djvu
4.1 The definitions......Page all_9140_to_00222.cpc0051.djvu
4.2 Examples......Page all_9140_to_00222.cpc0054.djvu
4.3 Convolutions with tempered distributions......Page all_9140_to_00222.cpc0060.djvu
4.4 Problems......Page all_9140_to_00222.cpc0062.djvu
5.1 The Laplace equation......Page all_9140_to_00222.cpc0064.djvu
5.2 The heat equation......Page all_9140_to_00222.cpc0068.djvu
5.3 The wave equation......Page all_9140_to_00222.cpc0070.djvu
5.4 Schrodinger's equation and quantum mechanics......Page all_9140_to_00222.cpc0075.djvu
5.5 Problems......Page all_9140_to_00222.cpc0077.djvu
6.1 The support of a distribution......Page all_9140_to_00222.cpc0081.djvu
6.2 Structure theorems......Page all_9140_to_00222.cpc0085.djvu
6.3 Distributions with point support......Page all_9140_to_00222.cpc0088.djvu
6.4 Positive distributions......Page all_9140_to_00222.cpc0091.djvu
6.5 Continuity of distribution......Page all_9140_to_00222.cpc0093.djvu
6.6 Approximation by test functions......Page all_9140_to_00222.cpc0100.djvu
6.7 Local theory of distributions......Page all_9140_to_00222.cpc0103.djvu
6.8 Problems......Page all_9140_to_00222.cpc0110.djvu
7.1 The Riemann-Lebesgue lemma......Page all_9140_to_00222.cpc0114.djvu
7.2 Paley-Wiener theorems......Page all_9140_to_00222.cpc0120.djvu
7.3 The Poisson summation formula......Page all_9140_to_00222.cpc0125.djvu
7.4 Probability measures and positive definite functions......Page all_9140_to_00222.cpc0130.djvu
7.5 The Heisenberg uncertainty principle......Page all_9140_to_00222.cpc0134.djvu
7.6 Hermite functions......Page all_9140_to_00222.cpc0139.djvu
7.7 Radial Fourier transforms and Bessel functions......Page all_9140_to_00222.cpc0143.djvu
7.8 Haar functions and wavelets......Page all_9140_to_00222.cpc0149.djvu
7.9 Problems......Page all_9140_to_00222.cpc0156.djvu
8.1 Sobolev inequalities......Page all_9140_to_00222.cpc0161.djvu
8.2 Sobolev spaces......Page all_9140_to_00222.cpc0170.djvu
8.3 Elliptic partial differential equations (constant coefficients)......Page all_9140_to_00222.cpc0174.djvu
8.4 Pseudodifferential operators......Page all_9140_to_00222.cpc0183.djvu
8.5 Hyperbolic operators......Page all_9140_to_00222.cpc0189.djvu
8.6 The wave front set......Page all_9140_to_00222.cpc0197.djvu
8.7 Microlocal analysis of singularities......Page all_9140_to_00222.cpc0206.djvu
8.8 Problems......Page all_9140_to_00222.cpc0210.djvu
Suggestions for Further Reading......Page all_9140_to_00222.cpc0215.djvu
Index......Page all_9140_to_00222.cpc0217.djvu
