nLab Fourier transform of distributions

Idea

The generalization of the concept of Fourier transform from suitable function to distributions.

Definition

Definition

(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

$$\array{ \mathcal{S}(\mathbb{R}^n) &\overset{\hat u}{\longrightarrow}& \mathbb{C} \\ f &\mapsto& \langle u, \hat f\rangle } \,,$$

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)

Properties

Proposition

The operation of Fourier transform of tempered distributions (def. ) induces a linear isomorphism of the space of temptered distributions with itself:

$$\widehat{(-)} \;\colon\; \mathcal{S}'(\mathbb{R}^n) \overset{\simeq}{\longrightarrow} \mathcal{S}'(\mathbb{R}^n)$$

(e.g. Melrose 03, corollary 1.1)

Proposition

(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

$$\hat u \in C^\infty(\mathbb{R}^n)$$

given by

$$\hat u(k) \;\coloneqq\; u\left( \exp(-i \langle (-), 2 \pi k \rangle) \right) \,,$$

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.)

Proposition

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:

$$\widehat {u_1 \star u_2} \;=\; \hat u_1 \hat u_2 \,.$$

(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)

Related concepts

• pullback of distributions

• product of distributions

• composition of distributions?

• extension of distributions

• Paley-Wiener-Schwartz theorem

References

• 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)