Existence and accuracy results for composite dilation wavelets

In this dissertation, we answer some existence questions for composite dilation wavelets generating orthonorinal bases for the space of square integrable functions defined on real Euclidean spaces of arbitrary dimension. We introduce an admissibility condition that determines which full rank lattices and which finite subgroups of the group of invertible matrices will combine to allow for the existence of a minimally supported frequency composite dilation scaling function. An admissibility condition for expanding matrices related to these lattices and groups determines when minimally supported frequency composite dilation wavelets may exist. We prove that these admissibility conditions are sufficient conditions for the existence of a multiresolution analysis (MRA), minimally supported frequency (MSF) composite dilation wavelet. We then show that every finite group of invertible matrices which has a fundamental region bounded by hyperplanes through the origin will admit an arbitrary lattice. For this large set of finite groups, we prove the existence of MRA, MSF composite dilation wavelets in every dimension. To improve applicability, we also prove the existence of such wavelets with a single wavelet generator. We answer similar questions for composite dilation wavelets generating Parseval frames rather than orthonormal bases. Using known accuracy results for several multivariate refinable functions, we examine composite dilation wavelets in the time domain in search of those with compact support. We establish the necessary equations to find compactly supported composite dilation wavelets with arbitrary levels of accuracy. We exhibit the first known compactly supported composite dilation wavelet with accuracy greater than the known Hater-type composite dilation wavelets.