A Guide to Localized Frames and Applications to Galerkin-like Representations of Operators

TL;DR

This paper provides a comprehensive survey of localized frames, exploring their properties and applications in representing operators through Galerkin-like schemes, with a focus on boundedness and invertibility conditions.

Contribution

It offers a detailed analysis of localized frames and their use in operator representation, including new conditions for boundedness and invertibility in Banach spaces.

Findings

Characterization of boundedness conditions for operators

Link between matrix and operator invertibility

Detailed properties of localized frames and Banach spaces

Abstract

This chapter offers a detailed survey on intrinsically localized frames and the corresponding matrix representation of operators. We re-investigate the properties of localized frames and the associated Banach spaces in full detail. We investigate the representation of operators using localized frames in a Galerkin-type scheme. We show how the boundedness and the invertibility of matrices and operators are linked and give some sufficient and necessary conditions for the boundedness of operators between the associated Banach spaces.