Skip to main content
Log in

Biorthogonal Multiwavelets on the Interval: Cubic Hermite Splines

  • Published:
Constructive Approximation Aims and scope

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]) .

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Author information

Authors and Affiliations

Authors

Additional information

February 2, 1998. Date revised: February 19, 1999. Date accepted: March 5, 1999.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dahmen, W., Han, B., Jia, RQ. et al. Biorthogonal Multiwavelets on the Interval: Cubic Hermite Splines. Constr. Approx. 16, 221–259 (2000). https://doi.org/10.1007/s003659910010

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s003659910010

Navigation