Interpolating Scaling Vectors and Multiwavelets in Rd

Karsten Koch

ISBN 978-3-8325-1489-1
180 pages, year of publication: 2007
price: 40.50 €

Within the last two decades, wavelet analysis has become a very powerful tool in applied mathematics. Wavelet algorithms have been successfully applied in signal analysis and compression as well as in numerical analysis, geophysics, meteorology and in many other fields. Usually, wavelets are constructed by means of a multiresolution analysis which, in turn, is generated by a scaling function. It is commonly known, however, that a scaling function fails to be interpolating while possessing a compact support and orthonormal integer translates. Hence the classical wavelet setting is somewhat restricted.

In this thesis, we intend to bypass this lack off flexibility by using multiwavelets and scaling vectors. Our main results are several recipes for the construction of interpolating scaling vectors with nice additional properties. First of all, we design univariate scaling vectors which are interpolating, orthonormal, and compactly supported simultaneously. As a next step, this univariate construction method is generalized to the multivariate case by utilizing expanding scaling matrices. In a third approach, we leave the orthonormal setting and focus on biorthogonal systems which possess certain symmetry properties. In addition to these construction methods, we address the problem of finding some multiwavelets and end up with an algorithm which leads to canonical multiwavelets corresponding to arbitrary interpolating scaling vectors. All our recipes are substantiated by various examples which indicate that the vector approach is capable of overcoming the restrictions of the scalar setting. Furthermore, we show that these results are suitable for application purposes as well.