Patrick Van Fleet

Center for Applied Mathematics, University of St. Thomas

Factoring Orthogonal Multiwavelet Transformations

Abstract

In this talk we will begin by giving a brief overview of basic scaling function/wavelet theory.  Central to this theory is Ingrid Daubechies' famous derivation of her family of orthogonal scaling functions.  These functions can be used to create a decomposition of L2(R) that has applications in a wide variety of areas.  The scaling function that generates this decomposition is (1) compactly supported, (2) suitably smooth, and, (3) along with its integer translates, forms an orthonormal basis for a subspace of L2(R).   In several applications, it is desirable to create a scaling function that is (4) symmetric or antisymmetric.  It turns out that a scaling function cannot be created to satisfy all four conditions.

We can generalize the above ideas to create a so-called scaling vector. It is then possible to create scaling vectors that do indeed satisfy all four properties listed in the preceding paragraph.  We will outline this generalization and then discuss the implementation of the theory by way of the discrete (multi)wavelet transformation.  We will provide some new results regarding the factorization of a certain class of orthogonal multiwavelet transformations.
Last updated by fass@amadeus.math.iit.edu  on 03/31/04