Exercise 36, § 1

Let $\sigma$ be a positive real number. We denote by $\mathscr{E}_{\sigma}$ the complex vector space of entire functions $f$ on $\mathbf{C}$ such that, for every real number $\varepsilon >0$, one has

$|f(z)|= O(e^{2\pi(\sigma+\varepsilon)|z|})$, when $|z| \rightarrow +\infty$.

a) Suppose $\sigma >0$. Let $g\in L^2([-\sigma , \sigma ])$. The function

belongs to $\mathscr{E}_{\sigma}$. The restriction of $f$ to $\mathbf{R}$ belongs to $L^2(\mathbf{R})$.

Let $f\in \mathscr{E}_{\sigma}$ such that the restriction of $f$ to $\mathbf{R}$ belongs to $L^2(\mathbf{R})$. Let $g\in L^2(\mathbf{R})$ be the Fourier transform of the restriction of $f$ to $\mathbf{R}$. The purpose of this exercise is to prove that the support of $g$ is contained in $[-\sigma , \sigma ]$; according to question a), this provides a characterization of the space $\mathscr{E}_{\sigma}$ (« Paley-Wiener theorem »).

b) For every real number $\theta$, the function

$h_{\theta}(z) =e^{-i\theta}\int_0^{+\infty}f(re^{-i\theta})$ exp($-2\pi zre^{-i\theta}$)$dr$

is holomorphic in the half-plane $H_{\theta}$ of $\mathbf{C}$ determined by the inequality

$\mathscr{R}(z)$ cos($\theta$ ) $+\mathscr{I}(z)$ sin($\theta$ )$> \sigma$.

c) If $\theta_1$ and $\theta_2$ are such that $|\theta_1-\theta_2|< \pi$, then $h_{\theta_1}|H_{\theta_1}\cap H_{\theta_2}=h_{\theta_2}|H_{\theta_2}\cap H_{\theta_1}$. (Apply Cauchy's formula.)

d) The function $h_0$ (resp. $h_{\pi}$) is holomorphic in the open set of the $z\in \mathbf{C}$ such that $\mathscr{R}(z)>0$ (resp. $\mathscr{R}(z)<0$).

e) Let $U =\mathbf{C}-i[-\sigma , \sigma ]$. There exists a unique holomorphic function $h$ on U such that $h(z) =h_0(z)$ if $\mathscr{R}(z)>0$ and $h(z) =h_{\pi}(z)$ if $\mathscr{R}(z)<0$.

f) When $x\rightarrow 0$ by values $>0$, the functions

converge towards $g$ in $L^2(\mathbf{R})$.

g) Conclude that $g$ is zero almost everywhere outside $[-\sigma , \sigma ]$.