The generalization of the concept of Fourier transform from suitable function to distributions.
(Fourier transform of tempered distributions)
For , let be a tempered distribution. Then its Fourier transform is defined by
where on the right is the ordinary Fourier transform of the function (an element of the Schwartz space ).
(e.g. Hörmander 90, def. 7.1.9)
The operation of Fourier transform of tempered distributions (def. ) induces a linear isomorphism of the space of temptered distributions with itself:
(e.g. Melrose 03, corollary 1.1)
(Fourier transform of compactly supported distributions)
If happens to be a compactly supported distribution, regarded as a tempered distribution, then its Fourier transform according to def. is a smooth function
given by
where denotes the canonical inner product on .
This is well-defined also on complex numbers, which makes it an entire holomorphic function (by the Paley-Wiener-Schwartz theorem), called the Fourier-Laplace transform.
(e.g. Hörmander 90, theorem 7.1.14)
(This plays a role for instance in the Paley-Wiener-Schwartz theorem.)
The Fourier transform (def. ) of the convolution of distributions of a compactly supported distribution with a tempered distribution is the product of distributions of their separate Fourier transforms:
(Here, by prop. , is just a smooth function, so that the product on the right is just that of a distribution with a function.)
(Hörmander 90, theorem 7.1.15)
composition of distributions?
Lars Hörmander, chapter VII of The analysis of linear partial differential operators, vol. I, Springer 1983, 1990
Richard Melrose, chapter 1 of Introduction to microlocal analysis, 2003 (pdf)
Sergiu Klainerman, chapter 5 of Lecture notes in analysis, 2011 (pdf)
Last revised on April 2, 2020 at 13:32:47. See the history of this page for a list of all contributions to it.