Abstract
The Daubechies wavelet based differentiation matrix will be constructed for periodic boundary conditions. It will be proved that this matrix displays the very important property of superconvergence. The relationship between Daubechies-based numerical methods and finite difference methods will be seen.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beylkin, G. On the Representation of Operators in Bases of Compactly Supported Wavelets (preprint).
Beylkin, G., Coifman, R., and Rokhin, V. (1991). Fast Wavelet Transforms and Numerical Algorithms I,Comm. Pure Appl. Math. 64, 141–184.
Daubechies, I. (1988). Orthonormal Basis of Compactly Supported Wavelets,Comm. Pure Appl. Math. 41, 909–996.
Davis, P. (1979).Circulant Matices, Wiley-Interscience.
Esteban, D., and Garland, C. (1977). Applications of Quadrature Mirror Filters to Split Band Voice Coding Schemes,Proc. Int'l. Conf. Acoustic Speech and Signal Proc.
Fairweather, G. (1978). Finite Element Galerkin Methods for Differential Equations, Marcel Dekker, pp. 43–49.
Mallat, S. (1989a). A Theory for Multiresolution Signal Decomposition: The Wavelet Representation,IEEE Trans. Pattern Anal. and Machine Intel. 11, 674–693.
Mallat, S. (1989b). Multiresolution Approximations and Wavelet Orthonormal Bases ofL 2(R),Trans. Amer. Math. Soc. 315(1), 69–87.
Strang, G. (1986).Introduction to Applied Mathematics, Wellesley-Cambridge Press, pp. 297–298.
Strang, G. (1989). Wavelets and Dilation Equations: A Brief Introduction,SIAM Review 31(4), 614–627.
Author information
Authors and Affiliations
Additional information
This research was supported by AFOSR Grant 90-0093, by DARPA grant N00014-91-4016, and by NSF grant DMS-9211820, in partial fulfillment of a Ph.D. in Applied Mathematics under the guidance of Professor David Gottlieb.
Rights and permissions
About this article
Cite this article
Jameson, L. On the wavelet based differentiation matrix. J Sci Comput 8, 267–305 (1993). https://doi.org/10.1007/BF01060934
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01060934