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.
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