Nehari’s Theorem for Convex Domain Hankel and Toeplitz Operators in Several Variables

We prove Nehari’s theorem for integral Hankel and Toeplitz operators on simple convex polytopes in several variables. A special case of the theorem, generalizing the boundedness criterion of the Hankel and Toeplitz operators on the Paley–Wiener space, reads as follows. Let $\Xi = (0,1)^d$ be a $d$-dimensional cube, and for a distribution $f$ on $2\Xi $, consider the Hankel operator $$\Gamma_f (g)(x)=\int_{\Xi} f(x+y) g(y) \, dy, \quad x \in\Xi.$$ Then $\Gamma _f$ extends to a bounded operator on $L^2(\Xi )$ if and only if there is a bounded function $b$ on ${{\mathbb{R}}}^d$ whose Fourier transform coincides with $f$ on $2\Xi $. This special case has an immediate application in matrix extension theory: every finite multilevel block Toeplitz matrix can be boundedly extended to an infinite multilevel block Toeplitz matrix. In particular, block Toeplitz operators with blocks that are themselves Toeplitz can be extended to bounded infinite block Toeplitz operators with Toeplitz blocks.