Wavelets and C*-algebras

Author: Peter John Wood

Wood, Peter John, 2003 Wavelets and C*-algebras, Flinders University, School of Informatics and Engineering

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Abstract

A wavelet is a function which is used to construct a specific type of orthonormal basis. We are interested in using C*-algebras and Hilbert C*-modules to study wavelets. A Hilbert C*-module is a generalisation of a Hilbert space for which the inner product takes its values in a C*-algebra instead of the complex numbers. We study wavelets in an arbitrary Hilbert space and construct some Hilbert C*-modules over a group C*-algebra which will be used to study the properties of wavelets. We study wavelets by constructing Hilbert C*-modules over C*-algebras generated by groups of translations. We shall examine how this construction works in both the Fourier and non-Fourier domains. We also make use of Hilbert C*-modules over the space of essentially bounded functions on tori. We shall use the Hilbert C*-modules mentioned above to study wavelet and scaling filters, the fast wavelet transform, and the cascade algorithm. We shall furthermore use Hilbert C*-modules over matrix C*-algebras to study multiwavelets.

Keywords: wavelet,filter,C*-algebra,Hilbert C*-module,cascade algorithm
Subject: Mathematics thesis

Thesis type: Doctor of Philosophy
Completed: 2003
School: School of Computer Science, Engineering and Mathematics
Supervisor: Peter Dodds