The discrete twofold Ellis-Gohberg inverse problem

TL;DR

This paper addresses a twofold inverse problem for orthogonal matrix functions in the Wiener class, extending scalar solutions to the matrix case and establishing conditions for existence and uniqueness of solutions.

Contribution

The paper generalizes the scalar Ellis-Gohberg inverse problem to matrix functions, providing invertibility criteria and analyzing special cases like matrix polynomials.

Findings

Invertibility condition based on Hankel and Toeplitz operators

Uniqueness of the solution under certain conditions

Analysis of special cases such as matrix polynomials

Abstract

In this paper a twofold inverse problem for orthogonal matrix functions in the Wiener class is considered. The scalar-valued version of this problem was solved by Ellis and Gohberg in 1992. Under reasonable conditions, the problem is reduced to an invertibility condition on an operator that is defined using the Hankel and Toeplitz operators associated to the Wiener class functions that comprise the data set of the inverse problem. It is also shown that in this case the solution is unique. Special attention is given to the case that the Hankel operator of the solution is a strict contraction and the case where the functions are matrix polynomials.