Quaternionic Wiener Algebras, Factorization and Applications

TL;DR

This paper extends Wiener algebras to the quaternionic setting, proving a Wiener-Levy theorem, a Wiener-Hopf factorization, and applying these results to solve quaternionic functional equations and rational matrix functions.

Contribution

It introduces quaternionic Wiener algebras, establishes their factorization properties, and connects these to the Riemann-Hilbert problem, advancing quaternionic analysis.

Findings

Proved a Wiener-Levy theorem for quaternionic Wiener algebras.

Established Wiener-Hopf factorization for quaternionic matrix-valued algebras.

Provided explicit formulas for quaternionic rational matrix function factorizations.

Abstract

We define an almost periodic extension of the Wiener algebras in the quaternionic setting and prove a Wiener-Levy type theorem for it, as well as extending the theorem to the matrix-valued case. We prove a Wiener-Hopf factorization theorem for the quaternionic matrix-valued Wiener algebras (discrete and continuous) and explore the connection to the Riemann-Hilbert problem in that setting. As applications, we characterize solvability of two classes of quaternionic functional equations and give an explicit formula for the canonical factorization of quaternionic rational matrix functions via realization.