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Constructive Approximation

, Volume 16, Issue 2, pp 221–259 | Cite as

Biorthogonal Multiwavelets on the Interval: Cubic Hermite Splines

  • W. Dahmen
  • B. Han
  • R.-Q. Jia
  • A. Kunoth
Article

Abstract.

Starting with Hermite cubic splines as the primal multigenerator, first a dual multigenerator on R is constructed that consists of continuous functions, has small support, and is exact of order 2. We then derive multiresolution sequences on the interval while retaining the polynomial exactness on the primal and dual sides. This guarantees moment conditions of the corresponding wavelets. The concept of stable completions [CDP] is then used to construct the corresponding primal and dual multiwavelets on the interval as follows. An appropriate variation of what is known as a hierarchical basis in finite element methods is shown to be an initial completion. This is then, in a second step, projected into the desired complements spanned by compactly supported biorthogonal multiwavelets. The masks of all multigenerators and multiwavelets are finite so that decomposition and reconstruction algorithms are simple and efficient. Furthermore, in addition to the Jackson estimates which follow from the exactness, one can also show Bernstein inequalities for the primal and dual multiresolutions. Consequently, sequence norms for the coefficients based on such multiwavelet expansions characterize Sobolev norms ||⋅|| Hs([0,1]) for s∈ (-0.824926,2.5) . In particular, the multiwavelets form Riesz bases for L 2 ([0,1]) .

Key words. Hermite cubic splines, Biorthogonal multigenerator, Multiwavelets, Interval, Stable completion. AMS Classification. 41A25, 42A38, 39B62. 

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Copyright information

© Springer-Verlag New York Inc. 2000

Authors and Affiliations

  • W. Dahmen
    • 1
  • B. Han
    • 2
  • R.-Q. Jia
    • 3
  • A. Kunoth
    • 1
  1. 1.Institut für Geometrie und Praktische Mathematik RWTH AachenAachenGermany
  2. 2.Program in Applied and Computational Mathematics Princeton University PrincetonUSA
  3. 3.Department of Mathematical SciencesUniversity of AlbertaEdmontonCanada

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