Title

Continuous Framings For Banach Spaces

Keywords

Continuous frame; Continuous framing; G-Köthe function space; Integral reconstruction

Abstract

The theory of discrete and continuous frames was introduced for the purpose of analyzing and reconstructing signals mainly in Hilbert spaces. However, in many interesting applications the analyzed space is usually a Banach space, and consequently the stable analysis/reconstruction schemes need to be investigated for general Banach spaces. Parallel to discrete Hilbert space frames, the theory of atomic decompositions, p-frames and framings have been introduced in the literature to address this problem. In this paper we focus on continuous frames and continuous framings (alternatively, integral reconstructions) for Banach spaces by the means of g-Köthe function spaces, in which the involved measure space is σ-finite, positive and complete. Necessary and sufficient conditions for a measurable function to be an Lρ-frame are obtained, and we obtain a decomposition result for the analysis operators of continuous frames in terms of simple Köthe-Bochner operators. As a byproduct we show that a Riesz type continuous frame doesn't exist unless the measure space is purely atomic. One of our main results shows that there is an intrinsic connection between continuous framings and g-Köthe function spaces.

Publication Date

8-15-2016

Publication Title

Journal of Functional Analysis

Volume

271

Issue

4

Number of Pages

992-1021

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1016/j.jfa.2016.05.010

Socpus ID

84969780634 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84969780634

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