Author: Wim Sweldens
Abstract: Wavelets are a new family of basis functions that can be used to approximate general functions. They combine powerful properties such as (bi)orthogonality, compact support, localization in time and frequency, and fast algorithms. This thesis investigates the use of wavelets in numerical analysis problems. In the first part we construct two basic tools, quadrature formulae and asymptotic error expansions. The former provides an easy way to calculate the wavelet coefficients, while the latter allows a simple comparison of different wavelet families. In the second part, we construct and study wavelets adapted to a weighted inner product. We show how one can use those wavelets for the rapid solution of ordinary differential equations. Finally, we study smooth local trigonometric functions, which can be seen as the Fourier transform of wavelets. We generalize their construction to the biorthogonal case, and show how to use them in data compression algorithms. This is illustrated with examples concerning image compression.
Status: PhD Thesis, Department of Computer Science, Katholieke Universiteit Leuven, Belgium, May 25, 1994.
BiBTeX entry:
@phdthesis {swe:phd,
author = {W. Sweldens},
title = {Construction and Applications of Wavelets in Numerical
Analysis},
school = {Department of Computer Science,
Katholieke Universiteit Leuven, Belgium},
year = {1994}
}
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