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

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.