Lecture 18 - fouriernew

Paley-Wiener theorem for distribution with compact support¶

Growth of Fourier-Laplace transform of compactly supported distributions¶

Let $T$ be a distribution with compact support, let $K$ be a compact set containing support of $K$ . Then

\begin{split} |\mathcal FT(z)| &= |\mathcal FT(x+iy)|\\ & = |(T(t),e^{-2\pi t(x+iy)})|\\ & \leq C e^{2\pi R|y|}\sum_{0\leq l\leq k}\sup_{x\in K}|\frac{d^{l}}{dx^l}e^{-2\pi i tx}|\\ &\leq C'e^{2\pi R|y|}(1+|z|)^k. \end{split}

(1)

where the second last row follows from previous lecture.

The Paley-Wiener theorem for distributions¶

Theorem. Let $U$ be an entire function satisfying $|U(z)|\leq C(1+|z|)^k e^{2\pi R|Im(z)|}$ for some integer $k>0$ . Then $U$ is the Fourier-Laplace transform of a distribution $T$ with compact support on $[-R,R]$ .

Idea of proof (informal)

The growth condition implies $U(x)$ is “moderate increasing” so gives a tempered distribution. Let $T = \mathcal F^{-1}U(x)$ . Then by the convolution theorem $\mathcal F(T*j_\epsilon) = \mathcal FU\cdot \mathcal Fj_\epsilon$ . Now since $j_\epsilon$ is compact supported in $[-\epsilon,\epsilon]$ , the Paley-Wiener theorem implies $\mathcal Fj_\epsilon$ has growth condition $|\mathcal Fj_\epsilon(z)|\leq C_k (1+|z|)^ke^{2\pi R|Im(z)|}$ for every positive integer $k$ . Combining with the growth condition for $U$ we see that the right hand side satisfies the grwoth condition of Paley-Wiener for $R+\epsilon$ . That shows $T*j_\epsilon$ should be a smooth function supported on $[-(R+\epsilon), R+\epsilon]$ . Since $T*j_\epsilon \to T$ weakly, this implies $T$ supported on $[-R,R]$ (to see the, choose any φ with support outside $[-R,R]$ , show that $(T,\varphi) = \lim_{\epsilon\to 0}(T*j_\epsilon,\varphi)\to 0$ ).

Proof. Let $V_\epsilon(z) = U(z)\mathcal Fj_\epsilon(z)$ . Since $j_\epsilon$ is supported on $[-\epsilon,\epsilon]$ , the Paley-Wiener theorem implies that $\mathcal F j_\epsilon(z)$ is an entire function satisfying the growth condition that

|\mathcal Fj_\epsilon(z)|\leq C_k (1+|z|)^{-k}e^{2\pi R|Im(z)|}

(2)

for every positive integer $k$ . Since $U$ is “moderate increasing”, $|U(z)|\leq C(1+|z|)^m e^{2\pi R|Im(z)|}$ for some integer $m>0$ . Overall, these growth estimates implies that $V_\epsilon(z)$ satisfies the growth condition of Paley-Wiener theorem for $R+\epsilon$ , i.e. $|V_\epsilon(z)|\leq C_{k'}(1+|z|)^{k'}e^{2\pi (R+\epsilon)|Im(z)|}$ for every positive integer $k'$ . The Paley-Wiener theorem implies that $V_\epsilon(z)$ is the Fourier-Laplace transform of some smooth function $v_\epsilon(x)$ supported on $[-(R+\epsilon),R+\epsilon]$ and $v_\epsilon(x) = \mathcal F^{-1}V_\epsilon(x) = \mathcal F^{-1}U(x) * \mathcal F^{-1}\mathcal Fj_\epsilon(x) = \mathcal F^{-1}U * j_\epsilon(x)$ . Note that the identity holds pointwise because they are both smooth functions and equal in sense of distribution (identity principle).

We have showed that $\mathrm{supp}(\mathcal F^{-1}U*j_\epsilon)\subset [-(R+\epsilon),R+\epsilon]$ . The support of convolution is equal to the Minkovski sum of supports the convolution elements, so this should imply $\mathrm{supp}\mathcal F^{-1}U \subset [-R,R]$ . To be precise, let $T = \mathcal F^{-1}U$ , for every $\varphi\in \mathcal D$ which is supported outside $[-R,R]$ , then eventually for some $\epsilon > 0$ , $\mathrm {supp}\varphi$ will be outside $[-(R+\epsilon),R+\epsilon]$ because $\mathrm{supp}\varphi$ is compact, and this implies $(T*j_\epsilon,\varphi) =0$ for small enough ε. In problem set 7 we have showed that $T*j_\epsilon \to T$ weakly, which means $\lim_{\epsilon\to 0} (T*j_\epsilon,\varphi)\to (T,\varphi)$ . This implies $(T,\varphi) =0$ and that $\mathrm{supp} T\subset [-R,R]$ .