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.

Thesis (PhD (Mathematics))--University of Stellenbosch, 2010. / Contains bibliography. / ENGLISH ABSTRACT: See full text for summary. / AFRIKAANSE OPSOMMING: Sien volteks vir opsomming

Wavelets are functions used to approximate data and can be traced back to several different areas, including seismic geology and quantum mechanics. Wavelets are applicable in many areas, including fingerprint and data compression, earthquake prediction, speech discrimination, and human vision. In this paper, we first give a brief history on the origins of wavelet theory. We will then discuss the work of Daubechies, whose construction of continuous, compactly supported scaling functions resulted in an explosion in the study of wavelets in the 1990's. These scaling functions allow for the construction of Daubechies' wavelets. Next, we shall use the algorithm to construct the Daubechies D4 scaling filters associated with the D4 scaling function. We then explore the Cascade Algorithm, which is a process that uses approximations to get possible representations for the D2N scaling function of Daubechies. Lastly, we will use the Cascade Algorithm to get a visual representation of the D4 scaling function.