## Distribution Theory: February-June 2007

This is a reading course aimed at master students in mathematics and mathematical physics.

### General information

Lecturer:   T.H. Koornwinder

Time and place: Friday, 9.00 (precisely) until 11.00 hour, room B.344, Nieuwe Achtergracht 166.
The last sessions will be on June 1, 9.30-11.00 hour in P.015A and on June 8, 9.00-11.00 in P.015B.

Credits: 6 EC

### Literature

J.J. Duistermaat and J.A.C. Kolk, Distributions: Theory and Applications, Birkhäuser
(a preliminary version has been made available to students). Students will also be pointed to a regularly updated list of errata.

#### Optional further reading

• G. Friedlander and M. Joshi, Introduction to the theory of distributions, Cambridge University Press, 1999, second printing (paperback).
• W. Rudin, Functional analysis, McGraw-Hill, 1973.
• L. Schwartz, Théorie des distributions, Hermann, Paris, 1966.

### Objectives

Distributions (or generalized functions) were introduced by the French mathematician Laurent Schwartz in 1945. The theory gives meaning to the (iterated) derivative of a continuous function on R or Rn, even if this function is non-differentiable. Distributions can be viewed as continuous linear functionals on a space of so-called test functions: the C^infinity functions of compact support. Operations on distributions, like the derivative, can be defined by their dual action on the space of test functions. The distributions form a linear space which contains the linear space of continuous functions, but also the linear space of complex measures, as subspaces. Dirac's delta "function" and its derivatives are famous examples of distributions.

The Fourier transform can be extended to a transform on the space of tempered distributions, a subspace of the space of all distributions. Here the rapidly decreasing C^infinity functions play the role of test functions.

Distributions are nowadays a standard tool in analysis, in the theory of PDE's, and in mathematical physics.

The course intends to provide a working knowledge of distribution theory, largely avoiding topological vector spaces, to develop Fourier analysis in the generalized setting of tempered distributions, and to give some tools for applying distribution theory to PDE's.

Contents:
test functions and distributions, differentiation, multiplication by smooth functions, tensor products of distributions, convolution, fundamental solutions, tempered distributions and Fourier transforms

### Form of teaching

Before each session students read one or two chapters from the book by Duistermaat and Kolk and prepare exercises from the book. During the session questions about the text will be answered, and some of the material will be further clarified or amplified. Next students in turn give solutions to exercises on the blackboard.

### Examination

A combination of exercises made in class, hand-in exercises and an oral examen.

### Links

See the paper Laurent Schwartz (1915-2002) in Notices Amer. Math. Soc., October 2003, Volume 50, Number 9, pp. 1072-1084.

to Tom Koornwinder's home page