Quadrature and Extremal Bandlimited Functions

Abstract

Let $\mathcal{A}(\delta)$ be the class of functions of exponential type $\delta>0$. We prove that for integrable $F\in\mathcal{A}(2\pi \delta)$ $\int_{-\infty}^\infty F(x) dx = \delta^{-1}\sum_{\xi\in\mathcal{T}_{\gamma,r}} (1-\frac{\gamma}{\pi(\xi^2+\gamma^2)+\gamma}) F(\delta^{-1}\xi)$, where $\mathcal{T}_{\gamma,r}$ is the set of zeros of $B_{\gamma,r}(z) = z\sin\pi(z+r) -\gamma\cos\pi(z+r)$. Let $a>(2\delta)^{-1}$. It is shown that for any Laguerre--Pólya entire function $E$ with $E(\pm a) =0$ there exist two integrable functions $G_-,G_+\in\mathcal{A}(2\pi \delta)$ such that for all real $x$ $E(x)[G_-(x) - \chi_{[-a,a]}(x)] \le 0$, $E(x)[G_+(x) - \chi_{[-a,a]}(x)] \ge 0.$ Combining these results we find the minimal value of $||S-T||_1$, where $S,T\in\mathcal{A}(2\pi \delta)$ satisfy $S(x)\le \chi_{[-a,a]}(x)\le T(x)$ for all real $x$. We determine extremal functions for which the minimal value is assumed. As an application we give an explicit expression for $C(\delta,\alpha) = \inf_{g\in\mathcal{A}_2(\delta)} \sup_{x\in[-\alpha,\alpha]} \frac{||g||_2^2}{|g(x)|^2}$, where $\mathcal{A}_2(\delta)$ is the set of square integrable functions in $\mathcal{A}(\delta)$. This constant occurs in work of Donoho and Logan regarding reconstruction of bandlimited functions.