Title: "Wavelets: Multiplicity theory, and group symmetries".
Abstract:
It is typically accepted that the Hilbert space where wavelet expansions are analyzed is L2(Rd). But there are other choices that naturally present themselves, both from the point of view of the theory and of its applications.
We will show that these other Hilbert spaces are natural when the familiar context, and formulation of scaling identity, and spaces of resolution/detail, is expanded somewhat. The applications of our results include:
-
both old and new constructions in geometric measure theory, -
in iterated function systems, and finally -
a new stability theorem for wavelets.
Our theory still begins with an analysis of subspaces which are invariant
under translations by the lattice Zd, and with an associated multiplicity
function. We will further sketch recent joint work with Larry Baggett, Kathy
Merrill, and Judy Packer where we show that our multiplicity analysis lets
us understand a family of multi-wavelets as a fibered bundle, with an
infinite-dimensional group acting naturally over each fiber. This
generalizes a formulation which originated in other joint research of the
speaker with Ola Bratteli, within the more traditional context of
multiresolutions. In this case, the multiplicity is one, and our group acts
transitively on the family of all wavelet filters.