Shea Yuen Cheuk. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 91-94). / Abstracts in English and Chinese. / Introduction --- p.5 / Chapter 1 --- Prelimaries --- p.9 / Chapter 1.1 --- Basic Notations --- p.9 / Chapter 1.2 --- Multiresolution Analysis --- p.12 / Chapter 1.3 --- Orthonormal Wavelets --- p.17 / Chapter 1.4 --- Theory of Frames --- p.24 / Chapter 2 --- Construction of Orthonormal Wavelets --- p.33 / Chapter 2.1 --- Compactly Supported Smooth Orthonormal Wavelet in R --- p.33 / Chapter 2.2 --- Compactly Supported Smooth Orthonormal Wavelet in R2 --- p.40 / Chapter 3 --- Wavelet Frames --- p.51 / Chapter 3.1 --- Basic Properties --- p.51 / Chapter 3.2 --- Dual Wavelet Frame --- p.56 / Chapter 3.3 --- Canonical Dual Frame --- p.66 / Chapter 3.4 --- Oversampling --- p.69 / Chapter 4 --- MRA-Based Wavelet Frames --- p.74 / Chapter 4.1 --- Definitions --- p.74 / Chapter 4.2 --- Tight Frames Constructed by MRA --- p.77 / Chapter 4.3 --- Approximation Order and Vanishing Moments for Wavelet Frames --- p.82 / Chapter 4.4 --- Construction of MRA-Based Wavelet Frames --- p.85 / Bibliography --- p.91

Thesis (Ph.D.)--Mathematics, Georgia Institute of Technology, 2008. / Committee Chair: Yang Wang; Committee Member: Christopher Heil; Committee Member: Doron Lubinsky; Committee Member: Guillermo Goldsztein; Committee Member: Steven W. McLaughlin

We consider two different problems in quantization theory. During the first part we discuss the so called Bennett's White Noise Hypothesis, introduced to study quantization errors of different schemes. Under this hypothesis, one assumes that the reconstruction errors of different channels can be considered as uniform, independent and identically distributed random variables. We prove that in the case of uniform quantization errors for frame expansions, this hypothesis is in fact false. Nevertheless, we also prove that in the case of fine quantization, the errors of different channels are asymptotically uncorrelated, validating, at least partially, results on the computation of the mean square error of reconstructions that were obtained through the assumption of Bennett's hypothesis. On the second part, we will introduced a new scalar quantization scheme, called a Beta Alpha Encoder. We analyze its robustness with respect to the quantizer imperfections. This scheme also induces a challenging dynamical system. We give partial results dealing with the ergodicity of this system.