WIAS Preprint No. 265, (1996)

Biorthogonal Spline-Wavelets on the Interval - Stability and Moment Conditions



Authors

  • Dahmen, Wolfgang
  • Kunoth, Angela
  • Urban, Karsten

2010 Mathematics Subject Classification

  • 15A12 35Q30 65F35 65N30 41A17 41A63

Keywords

  • Multiresolution analysis on the interval, biorthogonal wavelets, moment conditions, Riesz bases, discrete Sobolev norms

Abstract

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 as well as on the dual side 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 for L2([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.

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