Paley-Wiener-Schwartz theorem in nLab

Theorem

(Paley-Wiener-Schwartz theorem)

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

∀ξ∈ℂ n(|F(ξ)|≤C N(1+|ξ|) −Nexp(B|Im(ξ)|)). \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 ℝ n\mathbb{R}^n of order N∈ℕN \in \mathbb{N} is isomorphic via Fourier transform of distributions to those entire functions on ℂ n\mathbb{C}^n for which there exist positive real numbers B,C∈ℝ >0B, C \in \mathbb{R}_{\gt 0}

∀ξ∈ℂ n(|F(ξ)|≤C N(1+|ξ|) Nexp(B|Im(ξ)|)). \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..)

Proposition

(decay of Fourier transform of compactly supported functions)

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

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

|v^(ξ)|≤C N(1+|ξ|) −N. {\vert \hat v(\xi)\vert} \;\leq\; C_N \left( 1 + {\vert \xi\vert} \right)^{-N} \,.