Introduction to the Theory of Distributions

by F. G. Friedlander , M. Joshi

• ISBN13: 9780521649711
• ISBN10: 0521649714
• Edition: 2nd
• Format: Paperback
• Copyright: 1999-01-28
• Publisher: Cambridge University Press

Summary

The theory of distributions is an extension of classical analysis which has acquired a particular importance in the field of linear partial differential equations, as well as having many other applications, for example in harmonic analysis. Underlying it is the theory of topological vector spaces, but it is possible to give a systematic presentation without presupposing a knowledge, or using more than a bare minimum, of this. This book adopts this course and is based on graduate lectures given over a number of years. The prerequisites are few, but a reasonable degree of mathematical maturity is expected of the reader, as the treatment is rigorous throughout. From the outset the theory is developed in several variables, unlike most elementary texts; it is taken as far as such important topics as Schwartz kernels, the Paley-Wiener-Schwartz theorem and Sobolev spaces. In this second edition, the notion of the wavefront set of a distribution is introduced, which allows many operations on distributions to be extended to larger classes and gives much more precise understanding of the nature of the singularities of a distribution. This is done in an elementary fashion without using any involved theories. This account should therefore be useful to graduate students and research workers who are interested in the applications of analysis in mathematics and mathematical physics.

Table of Contents

Preface	ix
Introduction	1	(3)
Test functions and distributions	4	(13)
Some notations and definitions	4	(1)
Test functions	5	(2)
Distributions	7	(3)
Localization	10	(3)
Convergence of distributions	13	(4)
Exercises	15	(2)
Differentiation, and multiplication by smooth functions	17	(17)
The derivatives of a distribution	17	(1)
Some examples	18	(2)
A distribution obtained by analytic continuation	20	(2)
Primitives in D'(R)	22	(1)
Product of a distribution and a smooth function	23	(2)
Linear differential operators	25	(2)
Division in D'(R)	27	(2)
Duality	29	(5)
Exercises	30	(4)
Distributions with compact support	34	(6)
Continuous linear forms on C (X), and distributions with compact support	34	(2)
Distributions supported at the origin	36	(4)
Exercises	39	(1)
Tensor products	40	(10)
Test functions which depend on a parameter	40	(2)
Affine transformations	42	(2)
The tensor product of distributions	44	(6)
Exercises	48	(2)
Convolution	50	(18)
The convolution of two distributions	50	(3)
Regularization	53	(2)
Convolution of distributions with non-compact supports	55	(4)
Fundamental solutions of some differential operators	59	(9)
Exercises	65	(3)
Distribution kernels	68	(12)
Schwartz kernels and the kernel theorem	68	(5)
Regular kernels	73	(3)
Fundamental kernels of differential operators	76	(4)
Exercises	78	(2)
Coordinate transformations and pullbacks	80	(10)
Diffeomorphisms	80	(1)
The pullback of a distribution by a function	81	(4)
The wave equation on R4	85	(5)
Exercises	88	(2)
Tempered distributions and Fourier transforms	90	(24)
Introduction	90	(3)
Rapidly descreasing test functions	93	(3)
Tempered distributions	96	(5)
The convolution theorem	101	(3)
Poisson's summation formula, and periodic distributions	104	(4)
The elliptic regularity theorem	108	(6)
Exercises	110	(4)
Plancherel's theorem, and Sobolev spaces	114	(14)
Hilbert space	114	(2)
The Fourier transform on L2(Rn)	116	(4)
Sobolev spaces	120	(8)
Exercises	126	(2)
The Fourier-Laplace transform	128	(16)
Analytic functions of several complex variables	128	(2)
The Paley-Wiener-Schwartz theorem	130	(4)
An application to evolution operators	134	(5)
The Malgrange-Ehrenpreis theorem	139	(5)
Exercises	142	(2)
The calculus of wavefront sets	144	(18)
Definitions	144	(4)
Transformations of wavefront sets under elementary operations	148	(6)
Push-forwards and pull-backs	154	(3)
Wavefront sets and Schwartz kernels	157	(2)
Propagation of singularities	159	(3)
Exercises	160	(2)
Appendix: topological vector spaces	162	(8)
Bibliography	170	(1)
Notation	171	(2)
Index	173