Slanted matrices, Banach frames, and sampling

TL;DR

This paper explores the spectral properties of slanted matrices and applies these findings to improve understanding of Banach frames and irregular sampling, including a non-commutative Wiener's lemma for such matrices.

Contribution

It establishes that boundedness below in one ll^p space implies it in all ll^p spaces for a broad class of slanted matrices, advancing frame theory and sampling analysis.

Findings

Boundedness below in ll^p for some p implies it for all p.

New results for irregular sampling problems.

A version of non-commutative Wiener's lemma for slanted matrices.

Abstract

In this paper we present a rare combination of abstract results on the spectral properties of slanted matrices and some of their very specific applications to frame theory and sampling problems. We show that for a large class of slanted matrices boundedness below of the corresponding operator in $\ell^p$ for some $p$ implies boundedness below in $\ell^p$ for all $p$. We use the established resultto enrich our understanding of Banach frames and obtain new results for irregular sampling problems. We also present a version of a non-commutative Wiener's lemma for slanted matrices.