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   -> Volume 5, Issue 6

Preprint: Spline-Wavelets on the Interval
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kunoth@igpm.RWTH-Aachen.DE (Angela Kunoth)

PostPosted: Thu Aug 15, 1996 3:37 pm    
Subject: Preprint: Spline-Wavelets on the Interval
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Preprint: Spline-Wavelets on the Interval

The following preprint is available:

W. Dahmen, A. Kunoth, K. Urban
"Biorthogonal Spline-Wavelets on the Interval
- Stability and Moment Conditions"


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
$L_{2}([0,1])$ 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.

Angela Kunoth and Karsten Urban
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