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

Prakash A. Dabhi, Karishman B. Solanki

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.

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.
Paper Structure (13 sections, 31 theorems, 71 equations)

This paper contains 13 sections, 31 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weights
  4. Quasi-Beurling algebras
  5. Some known results
  6. Lemmas
  7. Wiener type theorem for countable limit of quasi-Beurling algebras
  8. One dimensional case
  9. Finite dimensional case
  10. Infinite dimensional case
  11. Hierarchy of inverse-closed algebras in quasi framework
  12. Maximizing the weight
  13. Non-quasi analogues and concluding remarks

Key Result

Theorem 1.1

For $d\in\mathbb{N}$ and $0<p\leq1$, $\ell^p_\omega(\mathbb{Z}^d)$ is inverse-closed if and only if $\omega$ is an admissible weight on $\mathbb{Z}^d$.

Theorems & Definitions (66)

  • Theorem 1.1: Wiener-Domar-Żelazko-GRS
  • Theorem 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Theorem 2.6
  • ...and 56 more