Continuous framings for Banach spaces

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.