Biorthogonal Spline Wavelets on the Interval—Stability and Moment Conditions

• Published in: Applied and computational harmonic analysis 1999-03, Vol.6 (2), p.132-196
• Main Authors: Dahmen, Wolfgang, Kunoth, Angela, Urban, Karsten
• Format: Article
• Language: English
• Subjects: biorthogonal wavelets; discrete Sobolev norms; moment conditions; multiresolution analysis on the interval; numerical stability; Riesz bases
• ISSN: 1063-5203; 1096-603X
• DOI: 10.1006/acha.1998.0247

This paper is concerned with the construction of biorthogonal multiresolution analyses on [0, 1] such that the corresponding wavelets realize any desired order of moment conditions throughout the interval. Our starting point is the family of biorthogonal pairs consisting of cardinal B-splines and compactly supported dual generators on R developed by Cohen, Daubechies, and Feauveau. In contrast to previous investigations we preserve the full degree of polynomial reproduction also for the dual multiresolution and prove in general that the corresponding modifications of dual generators near the end points of the interval still permit the biorthogonalization of the resulting bases. The subsequent construction of compactly supported biorthogonal wavelets is based on the concept of stable completions. As a first step we derive an initial decomposition of the spline spaces where the complement spaces between two successive levels are spanned by compactly supported splines which form uniformly stable bases on each level. As a second step these initial complements are then projected into the desired complements spanned by compactly supported biorthogonal wavelets. Since all generators and wavelets on the primal and the dual sides have finitely supported masks, the corresponding decomposition and reconstruction algorithms are simple and efficient. The desired number of vanishing moments is implied by the polynomial exactness of the dual multiresolution. Again due to the polynomial exactness the primal and dual spaces satisfy corresponding Jackson estimates. In addition, Bernstein inequalities can be shown to hold for a range of Sobolev norms depending on the regularity of the primal and dual wavelets. Then it follows from general principles that the wavelets form Riesz bases forL2([0, 1]) and that weighted sequence norms for the coefficients of such wavelet expansions characterize Sobolev spaces and their duals on [0, 1] within a range depending on the parameters in the Jackson and Bernstein estimates. We conclude by addressing several issues concerning numerical implementation. In particular, we test the quantitative stability properties of corresponding multiscale transformations, indicate strategies for improving them, and present some numerical experiments.