The generalization of the concept of Fourier transform from suitable function to distributions.
(Fourier transform of tempered distributions)
For $n \in \mathbb{N}$, let $u \in \mathcal{S}'(\mathbb{R}^n)$ be a tempered distribution. Then its Fourier transform $\hat u \in \mathcal{S}'(\mathbb{R}^n)$ is defined by
where on the right $\hat f$ is the ordinary Fourier transform of the function $f$ (an element of the Schwartz space $\mathcal{S}(\mathbb{R}^n)$).
(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 $u \in \mathcal{D}'(\mathbb{R}^n) \hookrightarrow \mathcal{E}'(\mathbb{R}^n)$ 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 $\langle -,-\rangle$ denotes the canonical inner product on $\mathbb{R}^n$.
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 $u_1 \in \mathcal{E}'$ with a tempered distribution $u_2 \in \mathcal{S}'$ is the product of distributions of their separate Fourier transforms:
(Here, by prop. , $\hat u_1$ 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 09:32:47. See the history of this page for a list of all contributions to it.