Frames of subspaces

By Peter G. Casazza, Gitta Kutyniok. 2003

434 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/dd585b06e27d231dbbe584388cc8acdad6c760c9

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In Applied Mathematics

In Computer Vision and Pattern Recognition

Citations in the corpus

Characterizations of g-frames and g-Riesz bases in Hilbert spaces (2008, methodology): “Since a frame of subspaces is a g-frame (see [9]), then from Theorem 2.4, we may obtain Theorem 3.12 in [ 11 ].”
The Structure of Minimizers of the Frame Potential on Fusion Frames (2008, background): “For an introduction to fusion frames for general Hilbert spaces, see [6, 11] or [20].”
G-frames and G-Riesz Bases (2005, background): “For example, bounded quasi-projectors [12, 13], frames of subspaces[2, 3], pseudo-frames[17], oblique frames[5, 10], and outer frames[1].”
On frames for countably generated Hilbert C*-modules (2006, background): “Therefore many statements from [4] can be extended to countably generated Hilbert K(H)-modules.”
On some identities and inequalities for frames in Hilbert spaces (2006, methodology): “Generalizations of frame theory were given in [1,12,27,30,31].”
G-FRAMES AND STABILITY OF G-FRAMES IN HILBERT SPACES (2008, background): “There are some generalizations of frames, for example bounded quasi-projectors [8] and frames of subspaces [ 3 ].”
On the duality of fusion frames (2007, neutral): “Generalizations of frame theory were give in [1,5,6,10,11,16,19].”
Fusion frames and distributed processing (2006, positive): “For the proof of the following result we refer to [11, Prop.”