Algebras of Almost Periodic Functions with Bohr-Fourier Spectrum in a Semigroup: Hermite Property and its Applications

TL;DR

This paper proves that certain algebras of almost periodic functions with spectra in a semigroup are Hermite rings, enabling advanced factorizations and solutions in multivariable harmonic analysis.

Contribution

It establishes the Hermite property for algebras of almost periodic functions with spectra in a semigroup, extending the algebraic framework for multivariable harmonic analysis.

Findings

The algebra of almost periodic functions with spectrum in a semigroup is an Hermite ring.

The Wiener algebra of such functions also has the Hermite property.

Applications include factorizations of matrix functions and solving the Toeplitz corona problem.

Abstract

It is proved that the unital Banach algebra of almost periodic functions of several variables with Bohr-Fourier spectrum in a given additive semigroup is an Hermite ring. The same property holds for the Wiener algebra of functions that in addition have absolutely convergent Bohr-Fourier series. As applications of the Hermite property of these algebras, we study factorizations of Wiener--Hopf type of rectangular matrix functions and the Toeplitz corona problem in the context of almost periodic functions of several variables.