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 $\phi$ we define a Hankel operator to be the densely defined operator: $$\widehat{H_{\phi}f}(x)=\int_{\Omega}\widehat{f}(y)\widehat{\phi}(x+y)dy,\text{ for $x\in\Omega$}.$$ 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.