Course on the Theory of Distributions: January - March 2002

docent                       T.H.  Koornwinder
Voorkennis                   1e en 2e jaarsvakken; van 3ejaars-vakken is
integratietheorie, functionaalanalyse, Fourier-analyse en eventueel
parti"ele differentiaalvergelijkingen gewenst.
Inhoud                       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 R^n, 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 will use the book by Friedlander & Joshi. It has the following
chapters, some of which may be skipped in the course (chapters 1,2,3,5 and 8 
will be certainly treated).
1. Test functions and distributions
2. Differentiation, and multiplication by smooth functions
3. Distributions with compact support
4. Tensor products
5. Convolution
6. Distribution kernels
7. Coordinate transformations and pullbacks
8. Tempered distributions and Fourier transforms
9. Plancherel's theorem, and Sobolev spaces
10. The Fourier-Laplace  transform
11. The calculus of wavefront sets
Onderwijsvorm                hoorcollege en werkcollege
Toetsing                     inleveropgaven en mondeling tentamen
Literatuur                   G. Friedlander & M. Joshi,
Introduction to the theory of distributions, Cambridge University Press, 1998,
second printing (paperback)
Opmerkingen                  College wordt desgewenst in het Engels gegeven:
theory of distributions.
College is ook interessant voor gevorderde studenten
theoretische natuurkunde en voor beginnende aio's.
Aanmelding                   verplicht

Note concerning the first author of the book to be used:
Dr F.G. Friedlander, FRS, who was elected a member of the London Mathematical Society on 19 June 1947, died on 20 May 2001, aged 83.

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