Abstract
In this paper a pair of wavelets are constructed on the basis of Hermite cubic splines. These wavelets are in C1 and supported on [−1,1]. Moreover, one wavelet is symmetric, and the other is antisymmetric. These spline wavelets are then adapted to the interval [0,1]. The construction of boundary wavelets is remarkably simple. Furthermore, global stability of the wavelet basis is established. The wavelet basis is used to solve the Sturm–Liouville equation with the Dirichlet boundary condition. Numerical examples are provided. The computational results demonstrate the advantage of the wavelet basis.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J.H. Bramble, J.E. Pasciak and J.C. Xu, Parallel multilevel preconditioners, Math. Comp. 55 (1990) 1–22.
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods (Springer, New York, 1994).
Z. Chen, C.A. Micchelli and Y. Xu, Discrete wavelet Petrov–Galerkin methods, Adv. Comput. Math. 16 (2002) 1–28.
C.K. Chui and E. Quak, Wavelets on a bounded interval, in: Numerical Methods in Approximation Theory, Vol. 9, eds. D. Braess and L.L. Schumaker (Birkhäuser, Basel, 1992) pp. 53–75.
C.K. Chui and J.Z. Wang, On compactly supported wavelets and a duality principle, Trans. Amer. Math. Soc. 330 (1992) 903–916.
P.G. Ciarlet, Introduction to Numerical Linear Algebra and Optimisation (Cambridge Univ. Press, Cambridge, 1989).
A. Cohen, I. Daubechies, and P. Vial, Wavelets on the interval and fast wavelet transforms, Appl. Comput. Harmonic Anal. 1 (1993) 54–81.
W. Dahmen, B. Han, R.Q. Jia and A. Kunoth, Biorthogonal multiwavelets on the interval: Cubic Hermite splines, Constr. Approx. 16 (2000) 221–259.
I. Daubechies, Ten Lectures on Wavelets (SIAM, Philadelphia, PA, 1992).
G. Donovan, J.S. Geronimo, D.P. Hardin and P.R. Massopust, Construction of orthogonal wavelets using fractal interpolation functions, SIAM J. Math. Anal. 27 (1996) 1158–1192.
C. Heil, G. Strang and V. Strela, Approximation by translates of refinable functions, Numer. Math. 73 (1996) 75–94.
R.Q. Jia and C.A. Micchelli, Using the refinement equations for the construction of pre-wavelets II: Powers of two, in: Curves and Surfaces, eds. P.J. Laurent, A. Le Méhauté and L.L. Schumaker (Academic Press, New York, 1991) pp. 209–246.
Y.J. Shen and W. Lin, A wavelet-Gelerkin method for a linear equation system with Hadamard integrals, manuscript.
J.Z. Wang, Cubic spline wavelet bases of Sobolev spaces and multilevel interpolation, Appl. Comput. Harmonic Anal. 3 (1996) 154–163.
J.-C. Xu and W.C. Shann, Galerkin-wavelet methods for two-point boundary value problems, Numer. Math. 63 (1992) 123–144.
Author information
Authors and Affiliations
Additional information
Communicated by Y. Xu
Dedicated to Dr. Charles A. Micchelli on the occasion of his 60th birthday
Mathematics subject classifications (2000)
42C40, 41A15, 65L60.
Research was supported in part by NSERC Canada under Grants # OGP 121336.
Rights and permissions
About this article
Cite this article
Jia, RQ., Liu, ST. Wavelet bases of Hermite cubic splines on the interval. Adv Comput Math 25, 23–39 (2006). https://doi.org/10.1007/s10444-003-7609-5
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10444-003-7609-5