Applications of Lax-Milgram theorem to problems in frame theory

TL;DR

This paper utilizes the Lax-Milgram theorem to analyze scalable frames in Hilbert spaces and introduces a novel finite-dimensional approach for approximating the inverse frame operator.

Contribution

It applies the Lax-Milgram theorem to characterize scalable frames and proposes a new finite-dimensional method for inverse frame operator approximation.

Findings

Characterization of scalable frames using Lax-Milgram theorem

Introduction of a new finite-dimensional approximation method

Potential applications in finite and infinite-dimensional Hilbert spaces

Abstract

We apply Lax-Milgram theorem to characterize scalable and piecewise scalable frame in finite and infinite-dimensional Hilbert spaces. We also introduce a method for approximating the inverse frame operator using finite-dimensional linear algebra which, to the best of our knowledge, is new in the literature.