Wiener type theorems for countable limits of quasi-Beurling algebras and maximizing results on weights

TL;DR

This paper extends Wiener type theorems to countable limits of quasi-Beurling algebras, establishing inverse-closedness and maximizing results for weights in both finite and infinite dimensional weighted settings.

Contribution

It introduces vector-valued Wiener theorems for quasi-Banach algebra limits and develops a hierarchy of inverse-closed weighted algebras with new maximizing results.

Findings

Quasi-Beurling algebras are inverse-closed in weighted settings.

Hierarchy of inverse-closed vector-valued algebras established.

Maximizing results on weights for nonadmissible Wiener's theorem derived.

Abstract

We establish the vector-valued Wiener type theorems for countable projective and inductive limits of quasi-Banach algebras in a weighted setting for both finite and infinite dimensional cases. As an application, we extend the notions of rapidly decreasing and exponentially decreasing sequence spaces using quasi-Beurling algebras and show that they are inverse-closed; and obtain a hierarchy of inverse-closed vector-valued algebras using weights. In addition, we derive maximizing results on weights for the nonadmissible weighted version of Wiener's theorem in both discrete and continuous cases.