nLab Paley-Wiener-Schwartz theorem

Idea

The Paley-Wiener-Schwartz theorem characterizes compactly supported smooth functions (bump functions) and more generally compactly supported distributions in terms of the decay property of their Fourier-Laplace transform (of distributions).

Conversely this means that for a general distribution those covectors along which its Fourier transform does not suitably decay detect the singular nature of the distribution (its failure to be represented by a smooth function) in a refined form which records not only the singular support (the point at which that covector is based), but also the “direction of propagation of singularities”. The collection of these covectors is called the wave front set of the distribution. The study of functional analysis with attention to the wave front sets is called microlocal analysis. This plays a central role in the definition of various operations on distributions, such as the pullback of distributions and the product of distributions.

Statement

Theorem

(Paley-Wiener-Schwartz theorem)

For $n \in \mathbb{N}$ the vector space $C^\infty_c(\mathbb{R}^n)$ of compactly supported smooth functions (bump functions) on Euclidean space $\mathbb{R}^n$ is (algebraically and topologically) isomorphic, via the Fourier transform, to the space of entire functions $F$ on $\mathbb{C}^n$ which satisfy the following estimate: there is a positive real number $B$ such that for every integer $N \gt 0$ there is a real number $C_N$ such that:

$$\underset{\xi \in \mathbb{C}^n}{\forall} \left( {\vert F(\xi) \vert} \le C_N (1 + {\vert \xi\vert })^{-N} \exp{ (B \; |Im(\xi)|)} \right) \,.$$

More generally, the space of compactly supported distributions on $\mathbb{R}^n$ of order $N \in \mathbb{N}$ is isomorphic via Fourier transform of distributions to those entire functions on $\mathbb{C}^n$ for which there exist positive real numbers $B, C \in \mathbb{R}_{\gt 0}$

$$\underset{\xi \in \mathbb{C}^n}{\forall} \left( {\vert F(\xi) \vert} \le C_N (1 + {\vert \xi\vert })^{N} \exp{ (B \; |Im(\xi)|)} \right) \,.$$

(Notice that the Fourier-Laplace transform of a compactly supported distribution is guaranteed to be an entire holomorphic function, by this prop..)

(e.g. Hoermander 90, theorem 7.3.1)

In fact

Proposition

(decay of Fourier transform of compactly supported functions)

A compactly supported distribution $u \in \mathcal{E}'(\mathbb{R}^n)$ is non-singular, hence given by a compactly supported function $b \in C^\infty_{cp}(\mathbb{R}^n)$ via $u(f) = \int b(x) f(x) dvol(x)$, precisely if its Fourier transform $\hat u$ (this def.) satisfies the following decay property:

For all $N \in \mathbb{N}$ there exists $C_N \in \mathbb{R}_+$ such that for all $\xi \in \mathbb{R}^n$ we have that the absolute value ${\vert \hat v(\xi)\vert}$ of the Fourier transform at that point is bounded by

$${\vert \hat v(\xi)\vert} \;\leq\; C_N \left( 1 + {\vert \xi\vert} \right)^{-N} \,.$$

(e.g. Hoermander 90, around (8.1.1))

Related entries

• Fourier inversion theorem

• Parseval's theorem

References

• Lars Hörmander, section 7.3 of The analysis of linear partial differential operators, vol. I, Springer 1983, 1990

See also

• Wikipedia, Paley-Wiener theorem