|
Organizers |
Generalized Frame Multiresolution Analysis of Abstract Hilbert spaces.
by
Manos Papadakis
Department of Mathematics, University of Houston, Houston, TX 77204-3476, USA
The Generalized frame multiresolution analysis of abstract Hilbert spaces is an MRA structure that encompasses orthogonal, biorthogonal both one dimensional and multidimensional MRA structures. Their core subspace, usually denoted by V0, has a frame which is produced by the action of an abelian group of unitary operators (called Translation group) on a countable set of vectors called set of frame multiscaling vectors. We prove that there exists a countable set of frame multiwavelet vectors associated with this structure in the sense that the action of the Translation group on this set produces a frame for the orthogonal complement of the core subspace with respect to its immediate succesor subspace (V1). We construct the set of the associated frame wavelet vectors by producing a pair of Generalized Quadratic Mirror Filters. Those filters are operator-valued functions contained in a Von Neumann algebra induced by the commutant of the Translation group. Finally we prove that Fast algorithms are associated with this type of MRAs. Also, we show how classical MRA structures and with the Frame MRAs of L2(R) are derived from these more general srtuctures. We conclude with some applications of this type of theory in the construction of non separable multidimensional scaling functions and their associated low pass filters, which emerged from our recent interactions with signal processing groups.
Date received: March 1, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadz-17.