Primbs, Miriam:

Stabile biorthogonale Spline-Waveletbasen auf dem Intervall

Duisburg (2006)
Dissertation / Fach: Mathematik
Fakultät für Mathematik
Abstract:
Based on the family of biorthogonal pairs of scaling functions consisting of cardinal B-splines and compactly supported dual generators on the whole real line, as presented by Cohen, Daubechies and Feauveau, we are concerned with the cnstruction of a new biorthogonal multiresolution analysis on the interval, such that the corresponding wavelet bases realize any desired order of moment condition on the interval. In contrast to previous approaches we choose the well-established Schoenberg spline basis on the interval with equidistant knots for primal multiresolution which has already been used by Chui and Quak to construct semiorthogonal spline bases on the interval. After giving an overview of the concrete construction, we discuss the cavorable properties of the constructed basis functions. The subsequent construction of the associated wavelets relies on the known method of stable completions, which has been presented by Dahmen, Kunoth and Urban as a helpful tool in constructing wavelet bases on the interval. Due to the fact, that we use all inner scaling functions, we use a high number of inner wavelets, so that the number of constructed boundary wavelets is very low, compared to former approaches. This is also true for the related wavelet basis with homogeneous or complementary boundary conditions. In view of applications, there are two interesting questions. Firstly, we investigate the condition number of the corresponding wavelet transforms. Secondly, we treat the stiffness matrix of the Laplace operator concerning to our basis and show, that its condition number is much better than the condition number in other approaches.