A Guide to Distribution Theory and Fourier Transforms

by Strichartz, Robert S.

• ISBN13: 9789812384300
• ISBN10: 9812384308
• Format: Paperback
• Copyright: 2003-09-01
• Publisher: World Scientific Pub Co Inc

Summary

This important 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 pseudodifferential 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.A Guide to Distribution Theory and Fourier Transforms can serve as a textbook for parts of a course on Applied Analysis or Methods of Mathematical Physics, and in fact it is used that way at Cornell.

Table of Contents

Preface	v
1 What are Distributions?	1	(11)
1.1 Generalized functions and test functions	1	(4)
1.2 Examples of distributions	5	(3)
1.3 What good are distributions?	8	(2)
1.4 Problems	10	(2)
2 The Calculus of Distributions	12	(16)
2.1 Functions as distributions	12	(2)
2.2 Operations on distributions	14	(4)
2.3 Adjoint identities	18	(2)
2.4 Consistency of derivatives	20	(2)
2.5 Distributional solutions of differential equations	22	(3)
2.6 Problems	25	(3)
3 Fourier Transforms	28	(18)
3.1 From Fourier series to Fourier integrals	28	(3)
3.2 The Schwartz class S	31	(1)
3.3 Properties of the Fourier transform on S	32	(6)
3.4 The Fourier inversion formula on S	38	(3)
3.5 The Fourier transform of a Gaussian	41	(2)
3.6 Problems	43	(3)
4 Fourier Transforms of Tempered Distributions	46	(14)
4.1 The definitions	46	(3)
4.2 Examples	49	(6)
4.3 Convolutions with tempered distributions	55	(2)
4.4 Problems	57	(3)
5 Solving Partial Differential Equations	60	(18)
5.1 The Laplace equation	60	(4)
5.2 The heat equation	64	(3)
5.3 The wave equation	67	(5)
5.4 Schrödinger's equation and quantum mechanics	72	(1)
5.5 Problems	73	(5)
6 The Structure of Distributions	78	(35)
6.1 The support of a distribution	78	(4)
6.2 Structure theorems	82	(3)
6.3 Distributions with point support	85	(3)
6.4 Positive distributions	88	(3)
6.5 Continuity of distribution	91	(7)
6.6 Approximation by test functions	98	(3)
6.7 Local theory of distributions	101	(2)
6.8 Distributions on spheres	103	(5)
6.9 Problems	108	(5)
7 Fourier Analysis	113	(49)
7.1 The Riemann-Lebesgue lemma	113	(6)
7.2 Paley-Wiener theorems	119	(6)
7.3 The Poisson summation formula	125	(5)
7.4 Probability measures and positive definite functions	130	(4)
7.5 The Heisenberg uncertainty principle	134	(5)
7.6 Hermite functions	139	(4)
7.7 Radial Fourier transforms and Bessel functions	143	(6)
7.8 Haar functions and wavelets	149	(8)
7.9 Problems	157	(5)
8 Sobolev Theory and Microlocal Analysis	162	(57)
8.1 Sobolev inequalities	162	(10)
8.2 Sobolev spaces	172	(4)
8.3 Elliptic partial differential equations (constant coefficients)	176	(9)
8.4 Pseudodifferential operators	185	(6)
8.5 Hyperbolic operators	191	(9)
8.6 The wave front set	200	(9)
8.7 Microlocal analysis of singularities	209	(5)
8.8 Problems	214	(5)
Suggestions for Further Reading	219	(2)
Index	221