The Bezout equation on the right half plane in a Wiener space setting

TL;DR

This paper solves the Bezout equation in the Wiener space on the right half plane, providing explicit solutions, including a minimal norm solution, and extends discrete case results to continuous Wiener functions.

Contribution

It introduces a Wiener space framework for the Bezout equation on the right half plane, describing all solutions explicitly and identifying a minimal norm solution.

Findings

Explicit description of all solutions in Wiener space

Identification of a minimal $H^2$-norm solution

Extension of discrete case results to continuous Wiener functions

Abstract

This paper deals with the Bezout equation $G(s)X(s)=I_m$, $\Re s \geq 0$, in the Wiener space of analytic matrix-valued functions on the right half plane. In particular, $G$ is an $m\times p$ matrix-valued analytic Wiener function, where $p\geq m$, and the solution $X$ is required to be an analytic Wiener function of size $p\times m$. The set of all solutions is described explicitly in terms of a $p\times p$ matrix-valued analytic Wiener function $Y$, which has an inverse in the analytic Wiener space, and an associated inner function $\Theta$ defined by $Y$ and the value of $G$ at infinity. Among the solutions, one is identified that minimizes the $H^2$-norm. A Wiener space version of Tolokonnikov's lemma plays an important role in the proofs. The results presented are natural analogs of those obtained for the discrete case in [11].