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.