Frames of subspaces

Abstract

One approach to ease the construction of frames is to first construct local components and then build a global frame from these. In this paper we will show that the study of the relation between a frame and its local components leads to the definition of a frame of subspaces. We introduce this new notion and prove that it provides us with the link we need. It will also turn out that frames of subspaces behave as a generalization of frames. In particular, we can define an analysis, a synthesis and a frame operator for a frame of subspaces, which even yield a reconstruction formula. Also concepts such as completeness, minimality, and exactness are introduced and investigated. We further study several constructions of frames of subspaces, and also of frames and Riesz frames using the theory of frames of subspaces. An important special case are harmonic frames of subspaces which generalize harmonic frames. We show that wavelet subspaces coming from multiresolution analysis belong to this class.1991 Mathematics Subject Classification. Primary 42C15; Secondary 46C99. Key words and phrases. Abstract frame theory, frame, harmonic frame, Hilbert space, resolution of the identity, Riesz basis, Riesz frame.

A decomposition theorem for frames and the Feichtinger Conjecture

“…Our idea of an ideal sequence is a sequence for which there exists a partition of its elements into finite sets such that the spans of the elements of those sets are mutually orthogonal; thus, properties of the sequence are completely determined by properties of its local components. This definition is inspired by a more general notion called fusion frames [9,10], which were designed to model distributed processing applications. This paper is organized as follows.…”

Some properties of alternate duals and approximate alternate duals of fusion frames

“…Fusion frame theory is a natural generalization of frame theory in separable Hilbert spaces, introduced by Casazza and Kutyniok in [4]. Fusion frames are applied to signal processing, image processing, sampling theory, filter banks, and a variety of applications that cannot be modeled by discrete frames [11,14].…”

“…It is clear that every Riesz fusion basis is a 1 -uniform fusion frame for H , and also a fusion frame is a Riesz basis if and only if it is a Riesz decomposition for H ; see [2,4].…”

Uniform recovery of fusion frame structured sparse signals

“…The collection of these subspaces may form a fusion frame (although this is not strictly required for our theory). Fusion frames generalize frames [11] and were first introduced in [9] under the name of 'frames of subspaces' (see also the survey [10]). They allow to analyze signals by projecting them onto multidimensional subspaces and for stable reconstruction from these projections.…”