A Functional Analytic Approach to Shift-Invariant Spaces, Frame Theory and Multiresolution Analysis
by
Manos Papadakis
Department of Mathematics, University of Houston, USA

First, we will present a unified approach to elementary frame theory of affine and Weyl-Heisenberg frames using an operator-theoretic approach; second, we will use the same techniques to study the structure of shift-invariant subspaces of abstract Hilbert spaces and apply our study in the cases of the L^2(R^n); third we will introduce a very general structure the Generalized Frame Multiresolution analysis, which incorporates most of the well known multiresolution structures. A prerequisite for these talks is the knowledge of the Hilbert space tensor product and of the concept of a Von Neumann algebra. We will present results on the density of Weyl-Heisenberg frames, the Daubechies frame identity and necessary and sufficient conditions for the density of these frames. We will give an operator-theoretic formulation of the Grammian that De Boor, De Vore, Ron and Shen introduced to study the structure of shift-invariant subspaces in one and multidimensions. Our approach is innovative and yields a better, more understanding of the concept of fibers, which are in a sense the ``atoms" of these subspaces. Finally, we show how we use these operator-theoretic techniques to study the Generalized Frame Multiresolution analysis, to derive quadrature mirror filters and frame multiwavelets in L^2(R^n) and other Hilbert spaces.

Date received: April 24, 2001

Copyright © 2001 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 # cagt-27.