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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