On the Kronecker and Caratheodory-Fejer theorems in several variables

TL;DR

This paper extends multivariate Kronecker and Carathéodory-Fejer theorems to continuous and discrete settings, characterizing symbols of finite-rank operators and structured matrices through elegant conditions and tensor algebra tools.

Contribution

It provides new characterizations of positive semidefinite symbols in multivariate settings and links continuous theorems to structured matrices via sampling and tensor algebra.

Findings

Continuous setting yields transparent if-and-only-if conditions.

Discretization leads to structured matrices with inherited properties.

New tensor algebra proof for block Toeplitz Carathéodory-Fejer theorem.

Abstract

Multivariate versions of the Kronecker theorem in the continuous multivariate setting has recently been published. These theorems characterize the symbols 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 semidefinite affects the characterization of the corresponding symbols, which we refer to as Carath\'eodory-Fejer type theorems. 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 objects can be discretized, giving rise to an interesting class of structured matrices that inherit these desirable properties from their continuous analogs. We describe how the continuous Kronecker theorem also applies to these structured matrices, given sufficient sampling. We also provide a new proof for the Carath\'eodory-Fejer theorem for block Toeplitz matrices, based on tools from tensor algebra.