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

Let $\mathrm{\Omega }$ be a nonempty, open and convex subset of ${\mathbb{R}}^n$. The Paley-Wiener space with respect to $\mathrm{\Omega }$ is defined to be the closed subspace of $L^2({\mathbb{R}}^n)$ of functions with Fourier transform supported in $2\mathrm{\Omega }$. For a tempered distribution $\phi$ we define a Hankel operator to be the densely defined operator: $$\widehat{H_{\phi }f}(x)={\int }_{\mathrm{\Omega }}\widehat{f}(y)\widehat{\phi }(x+y)dy,\text{ for }x\in \mathrm{\Omega }\mathrm{.}$$ We say that the Nehari theorem is true for $\mathrm{\Omega }$, 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.