On the Structure of Positive Semi-Definite Finite Rank General Domain Hankel and Toeplitz Operators in Several Variables

Abstract

Multivariate versions of the Kronecker theorem in the continuous multivariate setting has recently been published, that characterize the generating functions that give rise to finite rank multidimensional Hankel and Toeplitz type operators defined on general domains. In this paper we study how the additional assumption of positive semi-definite affects the characterization of the corresponding generating functions. We show that these theorems become particularly transparent in the continuous setting, by providing elegant if-and-only-if statements connecting the rank with sums of exponential functions. We also discuss how these operators can be discretized, giving rise to an interesting class of structured matrices that inherit desirable properties from their continuous analogs. In particular we describe how the continuous Kronecker theorem also applies to these structured matrices, given sufficient sampling. We also provide a new proof for the Carathéodory-Fejér theorem for block Toeplitz matrices, based on tools from tensor algebra.