Author: Peter John Wood
Wood, Peter John, 2003 Wavelets and C*-algebras, Flinders University, School of Informatics and Engineering
This electronic version is made publicly available by Flinders University in accordance with its open access policy for student theses. Copyright in this thesis remains with the author. This thesis may incorporate third party material which has been used by the author pursuant to Fair Dealing exceptions. If you are the owner of any included third party copyright material and/or you believe that any material has been made available without permission of the copyright owner please contact firstname.lastname@example.org with the details.
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
School: School of Computer Science, Engineering and Mathematics
Supervisor: Peter Dodds