Localized frames without inequalities

TL;DR

This paper establishes inequality-free conditions for frame properties of localized vector families in Hilbert spaces, with applications to shift-invariant spaces and sampling stability.

Contribution

It introduces nine equivalent conditions for frames without inequalities, expanding understanding of localized frames and their operators.

Findings

Equivalence of frame property and nine inequality-free conditions

Application to shift-invariant spaces for sampling stability

New criteria for stable sampling sets

Abstract

We consider countable families of vectors in a separable Hilbert space, which are mutually localized with respect to a fixed localized Riesz basis. We prove the equivalence of the frame property and nine conditions that do not involve any inequalities. This is done by studying the properties of their frame-related operators on the co-orbit spaces generated by the reference Riesz basis. We apply our main result to the setting of shift-invariant spaces and obtain new conditions for stable sets of sampling.