On the failure of the Nehari Theorem for Paley-Wiener spaces

Abstract

Let $\Omega$ be a nonempty, open and convex subset of ${\mathbb{R}}^n$. The Paley-Wiener space with respect to $\Omega$ is defined to be the closed subspace of $L^2({\mathbb{R}}^n)$ of functions with Fourier transform supported in $2\Omega$. For a tempered distribution $\varphi$ we define a Hankel operator to be the densely defined operator: $$\widehat{H_{\varphi }f}(x)={\int }_{\Omega }\hat{f}(y)\hat{\varphi }(x+y)dy,\text{ for x∈Ω}.$$ We say that the Nehari theorem is true for $\Omega$, if every bounded Hankel operator is generated by a bounded function. In this paper we prove that the Nehari theorem fails for any convex set in ${\mathbb{R}}^n$ that has infinitely many extreme points. In particular, it fails for all convex bounded sets which are not polytopes. Furthermore, in the setting of $R^2$, it fails for all non-polyhedral sets, bounded or unbounded.