Skip to main content
Log in

Wavelet-Galerkin methods: An adapted biorthogonal wavelet basis

  • Published:
Constructive Approximation Aims and scope

Abstract

In this paper we construct a compactly supported biorthogonal wavelet basis adapted to some simple differential operators. Moreover, we estimate the condition numbers of the corresponding stiffness matrices.

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

References

  1. A. S. Cavaretta, W. Dahmen, C. A. Micchelli (1991): Stationary Subdivision. Memoirs of the American Mathematical Society, vol. 93 Providence, RI: American Mathematical Society, No. 453.

    Google Scholar 

  2. C. Chui, I. Wang (1992):On compactly supported spline wavelets and a duality principle Trans. Amer. Math. Soc.,330:903–912.

    Google Scholar 

  3. C. Chui, I. Wang (1992):A general framework of compactly supported splines and wavelets. J. Approx. Theory,71:263–304.

    Google Scholar 

  4. A. Cohen, I. Daubechies, J. C. Feauveau (to appear):Biorthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math.

  5. W. Dahmen, A. Kunoth (to appear):Multilevel preconditioning. Numer. Math.

  6. W. Dahmen, C. A. Micchelli (1990):On stationary subdivision and the construction of compactly supported wavelets. In: Multivariate Approximation and Interpolation (W. Haussmann, K. Jetter, eds.). ISNM 94. Basel: Birkhauser-Verlag, pp. 69–89.

    Google Scholar 

  7. W. Dahmen, C. A. Micchelli (to appear):Dual wavelet expansions for general scalings

  8. I. Daubechies (1987):Orthonormal bases of wavelets with compact support. Comm. Pure Appl. Math.,41:909–996.

    Google Scholar 

  9. T. Eirola (1992):Sobolov characterization of solutions of dilation equations. SIAM J. Math. Anal.,23(4):1015–1030.

    Google Scholar 

  10. R. Glowinski, W. Lawton, M. Ravachol, E. Tenenbaum (1989):Wavelet solutions of linear and nonlinear elliptic, parabolic and hyperbolic problems in one space dimension. Preprint. Cambridge, MA: Aware.

    Google Scholar 

  11. S. Jaffard (1990):Wavelet methods for fast resolution of elliptic problems. In: Méthodes d'ondelettes pour la résolution d'équations aux dérivées partielles (S. Jaffard, ed.). LAMM Report.

  12. R. Q. Jia, C. A. Micchelli (1991):Using the refinement equations for the construction of pre-wavelets II: Powers of two. In: Curves and Surfaces (P. J. Laurent, A. Le Méhauté, L. L. Schumaker, eds.). New York: Academic Press.

    Google Scholar 

  13. R. A. Lorentz, W. Madych (to appear): Spline Wavelets for Ordinary Differential Equations. GmD-Report 1990.

  14. S. Mallat (1989):Multiresolution approximation and wavelet orthonormal bases of L 2. Trans. Amer. Math. Soc.,315:69–88.

    Google Scholar 

  15. Y. Meyer (1990): Ondelettes et Opérateurs I. Paris: Hermann.

    Google Scholar 

  16. H. Yserentant (1986):On the multilevel splitting of finite element spaces. Numer. Math.,49:379–412.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by Ronald A. DeVore.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dahlke, S., Weinreich, I. Wavelet-Galerkin methods: An adapted biorthogonal wavelet basis. Constr. Approx 9, 237–262 (1993). https://doi.org/10.1007/BF01198005

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01198005

AMS classification

Key words and phrases

Navigation