Taylor Matyasz

Western Washington University
, BH 419

Abstract

Wavelets: Mallat’s Theorem and Applications

One of the best features of a vector space is the idea of a basis, especiallyorthonormal bases. Every vector space has one and they are incrediblyuseful for both understanding and representing otherwise complicatedelements. In the finite dimensional case any collection of vectors that isboth orthonormal and spans the space is acceptable, no matter how crazythe vectors may look, since it is easy to understand a finite collection ofvectors. However when you extend to infinite dimensions, this is no longerthe case. If we wish to understand a vector space with an infinite basis, weneed some concise method of describing all of the vectors. Of particularconcern is the infinite dimensional space known asL2(R)and Mallat’stheorem will give us a way to construct a orthonormal basis that can bewritten as a collection of translations and dilations of a single vector, giventhat we already have a structure called a Multiresolution Analysis (orMRA).

Time permitting, we will also discuss the idea of wavelets as applied tonumerical methods and image compression.