nLab
Fourier transform of distributions

Contents

Idea

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

Definition

Definition

(Fourier transform of tempered distributions)

For nn \in \mathbb{N}, let u𝒮( n)u \in \mathcal{S}'(\mathbb{R}^n) be a tempered distribution. Then its Fourier transform u^𝒮( n)\hat u \in \mathcal{S}'(\mathbb{R}^n) is defined by

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

where on the right f^\hat f is the ordinary Fourier transform of the function ff (an element of the Schwartz space 𝒮( n)\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:

()^:𝒮( n)𝒮( n) \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 u𝒟( n)( n)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

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

given by

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

where ,\langle -,-\rangle denotes the canonical inner product on n\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 1u_1 \in \mathcal{E}' with a tempered distribution u 2𝒮u_2 \in \mathcal{S}' is the product of distributions of their separate Foruier transforms:

u 1u 2^=u^ 1u^ 2. \widehat {u_1 \star u_2} \;=\; \hat u_1 \hat u_2 \,.

(Here, by prop. , u^ 1\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)

References

Last revised on November 15, 2017 at 06:57:10. See the history of this page for a list of all contributions to it.