# Multiresolution analysis on non-abelian locally compact groups

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1999-01-01

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Doctoral

## Abstract

Wavelets are a relatively new mathematics. They have generated a tremendous interests in both theoretical and applied areas. Multiresolution analysis (MRA) is a important mathematical tool because it provides a natural framework for the understanding and constructing wavelets. Over past few years, almost all research work has been restricted on the space 'L' 2(R'd'), where R 'd' is the 'd'-dimensional Euclidean space that is a abelian group. As to non-abelian group, very few people have investigated MRA in this case. In this thesis, we plan on extending MRA to the setting of non-abelian locally compact groups. It can be seen as one initial step towards wavelet theory of non-abelian groups. Our motivation for the development of MRA for non-abelian groups comes from Heisenberg groups. The main contributions of the thesis are following: (1) We create a new term, called scalable groups, for a special class of groups. MRA can only be set up for the class of scalable groups. We approximately identify the class ofscalable groups out of second countable, type I, unimodular locally compact groups. (2) For a scalable group G, we formulate the definition of MRA for 'L' 2('G') by using the information exposed from the MRA of 'L'2 (R'd'). Generally speaking, the way to construct an MRA is to start with a refinable function [straight phi] with orthogonal shifts. Then form the central space ' V'0 which is a closed linear span of the shifts of [straight phi]. Finally, obtain a sequence of nested subspaces 'Vj' := [sigma]'j''V'0 by using the central space 'V'0 and the unitary operator [sigma]. There are three things in MRA that mainly concerned us, that is, the density of the union and the triviality of the intersection of the nested sequence of closed subspaces and the existence of refinable functions. We set up the union density and intersection triviality theorems and other related things for scalable groups. The intersection triviality property is a direct consequence of other conditions of MRA. To get the union density property, we have to generalize the concept of the support of the Fourier transform. The new concepts, such as "strongly supported", left nonzero divisor in 'L'2('G'), and automorphism-absorbing subset of 'G', arise in this generalization. These new ideas will enrich the original thoughts about the classical MRA. As to refinability, it depends very much on the individual function [straight phi]. We prove that the refinable functions are present for general scalable groups as long as self-similar tiles are present. (3) We provide a very interesting concrete example for our theory using Heisenberg groups. An MRA on Heisenberg groups is set up. We investigate the existence of scaling functions for the Heisenberg groups. These scaling functions are related to certain self-similar tilings of H'd', that is, the corresponding scaling functions are characteristic functions of appropriate sets. We generalize the construction of Strichartz's self-similar tiles to a more general case. We also obtain a theorem which says that there are 22d+2-1 orthonormal wavelets for Heisenberg groups.

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Doctor of Philosophy (Ph.D.)

## Department

Mathematics and Statistics

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Mathematics and Statistics