Exercise 56, § 1

We put

$k(z) =($ sin($\pi z\pi z$))$^2$

if $z\in \mathbf{C}-\mathbf{Z}$ and $k(z) = 1$ if $z\in \mathbf{Z}$. The function $k$ belongs to the space $\mathscr{E}_{1/2}$ (exercise 36).

(a) For every complex number $z\in \mathbf{C}-\mathbf{Z}$, we have

$\sum\frac{1}{(z-n)^2}=($sin($\pi \pi z$))$^2$

$n\in \mathbf{Z}$

(apply the Poisson formula).

(b) For $z\in \mathbf{C}-\mathbf{Z}$, we define

$h(z) =($sin($\pi \pi z$))$^2\{(\sum_{n\in\mathbf{Z}}(z$s($-nn$))$^2+\frac{2}{z}\}$,

where s($n$) denotes the sign function. The function $h$ extends to an entire function belonging to the space $\mathscr{E}_2$. We put $b=h+k$.

(c) For every $x\in \mathbf{R}$, we have $1-k(x)\leqslant h(x)\leqslant 1$.

(d) For every $x\in \mathbf{R}$, we have $|$s($x$)$-h(x)|\leqslant k(x)$.

(e) We have $b\geqslant$ s ; the function $b-$ s is integrable on $\mathbf{R}$ and satisfies $\int(b-$ s) = 1. f) Let $\alpha < \beta$ be real numbers and let $\varphi$ be the characteristic function of the interval $[\alpha , \beta ]$. The function $f_{\alpha ,\beta}(z) =\frac{1}{2}(b(\beta -z) +b(z-\alpha ))$ satisfies $\varphi \leqslant f_{\alpha ,\beta}$ and $\int_{\mathbf{R}}(f_{\alpha ,\beta}-\varphi ) = 1$. g) There exists an entire function $\sigma$ such that $f_{\alpha ,\beta}(x) =|\sigma (x)|^2$ for every $x\in \mathbf{R}$. The Fourier transform of the restriction of $\sigma$ to $\mathbf{R}$ is supported in $[-1/2,1/2]$.