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Fourier Analysis and distribution theory
| Credits |
8 credit points |
| Instructors |
Stolk, C.C. (Universiteit van Amsterdam), Wiegerinck, J.J.O.O. (Universiteit van Amsterdam) |
| E-mail |
C.C.Stolk@uva.nl, J.J.O.O.Wiegerinck@uva.nl |
| Description |
In Fourier analysis one studies functions on R, Rn, or more general spaces by writing them as linear superpositions of elementary functions. This is very clear in the situation of periodic functions. These may often be expressed as a linear superpositions of exponentials with the same period. It is interesting to study the connection between functions and their Fourier transform for its own sake, but it has turned out that Fourier analysis is a powerful tool to handle problems in differential equations, signal analysis etcetera. When one studies the classes of functions that permit Fourier analysis, it soon becomes clear that one should study a larger class of objects, the so called distributions or generalized functions. In this course all these aspects will be dealt with to some extent. |
| Organization |
Tentatively lectures from 10-12, exercises 12-13. Instruction First weeks, Usually Wiegerinck, Last weeks Wiegerinck and Stolk will vary regularly. |
| Examination |
Take home exercises, in class presentation, and or oral exam. |
| Literature |
- Compulsory!, Robert S. Strichartz: A guide to distribution Theory and Fourier transforms, World Scientific, ISBN-13 978-981-238-430-0.
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| Prerequisites |
Bachelor mathematics, preferably with linear analysis or Fourier analysis. |
| Remarks |
Contents: Classical L2-theory of Fourier series. Convergence problems, periodic distributions and their Fourier theory, Distributions in Rn, Fourier transform, L2-theory, Schwartz class and tempered distributions, Fourier transforms of distributions, Sobolev space interpolation, microlocal theory, applications to differential operators, wavelets, lacunary series. See also: http://staff.science.uva.nl/~janwieg/edu/Fourier/ |
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