  A wavelet is a special case of a vector in a separable Hilbert space that
generates a basis under the action of a collection, or system, of unitary
operators. We will describe the operator-interpolation approach to wavelet
theory using the local commutant of a system. This is really an abstract
application of the theory of operator algebras to wavelet theory. The concrete
applications of this method include results obtained using specially
constructed families of wavelet sets. A frame is a sequence of vectors in a
Hilbert space which is a compression of a basis for a larger space. This is not
the usual definition in the frame literature, but it is easily equivalent to
the usual definition. Because of this compression relationship between frames
and bases, the unitary system approach to wavelets (and more generally:
wandering vectors) is perfectly adaptable to frame theory. The use of the local
commutant is along the same lines as in the wavelet theory. Finally, we discuss
constructions of frames with special properties using targeted decompositions
of positive operators, and related problems.
