  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.
