Daubechies wavelets on intervals with application to BVPs. (English) Zbl 1099.65146

Summary: In this paper, Daubechies wavelets on intervals are investigated. An analytic technique for evaluating various types of integrals containing the scaling functions is proposed; they are compared with classical techniques. Finally, these results are applied to two-point boundary value problems.


65T60 Numerical methods for wavelets
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
34B05 Linear boundary value problems for ordinary differential equations
Full Text: DOI EuDML


[1] J. J. Benedetto, M. W. Frazier: Wavelets: Mathematics and Applications. Studies in Advanced Mathematics. CRC Press, Boca Raton, 1994. · Zbl 0840.00013
[2] A. Cohen: Wavelet Methods in Numerical Analysis. Handbook of Numerical Analysis, Vol. 7. P. G. Ciarlet at al. (eds.), North-Holland/Elsevier, Amsterdam, 2000, pp. 417-711. · Zbl 0976.65124
[3] I. Daubechies: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41 (1988), 909-996. · Zbl 0644.42026
[4] I. Daubechies: Ten Lectures on Wavelets. SIAM Publ., Philadelphia, 1992. · Zbl 0776.42018
[5] R. J. Duffin, A. C. Schaeffer: A class of nonharmonic Fourier series. Trans. Amer. Math. Soc. 72 (1952), 341-366. · Zbl 0049.32401
[6] V. Finěk: Daubechies wavelets and two-point boundary value problems. Preprint, TU Dresden, 2001.
[7] R. Glowinski, W. Lawton, M. Ravachol, and E. Tenenbaum: Wavelet solution of linear and nonlinear elliptic, parabolic and hyperbolic problems in one space dimension. Computing Methods in Applied Sciences and Engineering, Proc. 9th Int. Conf. Paris, 1990, pp. 55-120. · Zbl 0799.65109
[8] Ch. Grossmann, H.-G. Roos: Numerik partieller Differentialgleichungen, 2. edition. Teubner, Stuttgart, 1994.
[9] C. Heil: Wavelets and frames, Signal processing, Part I: Signal processing theory. Proc. Lect, , Minneapolis, 1988.
[10] A. Kunoth: Wavelet Methods-Elliptic Boundary Value Problems and Control Problems. Advances in Numerical Mathematics. Teubner, Stuttgart, 2001. · Zbl 1011.65080
[11] W. Lawton: Necessary and sufficient conditions for constructing orthonormal wavelet bases. J. Math. Phys. 32 (1991). · Zbl 0757.46012
[12] A. K. Louis, P. Maas, A. Rieder: Wavelets: Theorie und Anwendungen. Teubner, Stuttgart, 1994.
[13] Y. Meyer: Ondelettes et Opérateurs I-Ondelettes. Hermann Press, Paris, 1990, · Zbl 0694.41037
[14] Y. Meyer: Ondelettes sur l’intervalle. Rev. Math. Iberoamer. 7 (1991), 115-133. · Zbl 0753.42015
[15] Z.-Ch. Shann, J.-Ch. Yan: Quadratures involving polynomials and Daubechies’ wavelets. Preprint, National Central University, Chung-Li, Taiwan, R.O.C., April)
[16] W.-Ch. Shann, J.-Ch. Xu: Galerkin-wavelet methods for two-point boundary value problems. Numer. Math. 63 (1992), 123-144. · Zbl 0771.65050
[17] W. Sweldens, R. Piessens: Quadrature formulae and asymptotic error expansions for wavelet approximation of smooth functions. SIAM J. Numer. Anal. 31 (1994), 1240-1264. · Zbl 0822.65013
[18] P. Wojtaszczyk: A Mathematical Introduction to Wavelets. Cambridge University Press, Cambridge, 1997. · Zbl 0865.42026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.