, Volume 25, Issue 1–3, pp 23–39

# Wavelet bases of Hermite cubic splines on the interval

• Rong-Qing Jia
• Song-Tao Liu
Article

## 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.

## Keywords

wavelets on the interval Hermite cubic splines numerical solutions of differential equations

## References

1. [1]
J.H. Bramble, J.E. Pasciak and J.C. Xu, Parallel multilevel preconditioners, Math. Comp. 55 (1990) 1–22.
2. [2]
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods (Springer, New York, 1994). Google Scholar
3. [3]
Z. Chen, C.A. Micchelli and Y. Xu, Discrete wavelet Petrov–Galerkin methods, Adv. Comput. Math. 16 (2002) 1–28.
4. [4]
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. Google Scholar
5. [5]
C.K. Chui and J.Z. Wang, On compactly supported wavelets and a duality principle, Trans. Amer. Math. Soc. 330 (1992) 903–916.
6. [6]
P.G. Ciarlet, Introduction to Numerical Linear Algebra and Optimisation (Cambridge Univ. Press, Cambridge, 1989). Google Scholar
7. [7]
A. Cohen, I. Daubechies, and P. Vial, Wavelets on the interval and fast wavelet transforms, Appl. Comput. Harmonic Anal. 1 (1993) 54–81.
8. [8]
W. Dahmen, B. Han, R.Q. Jia and A. Kunoth, Biorthogonal multiwavelets on the interval: Cubic Hermite splines, Constr. Approx. 16 (2000) 221–259.
9. [9]
I. Daubechies, Ten Lectures on Wavelets (SIAM, Philadelphia, PA, 1992). Google Scholar
10. [10]
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.
11. [11]
C. Heil, G. Strang and V. Strela, Approximation by translates of refinable functions, Numer. Math. 73 (1996) 75–94.
12. [12]
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. Google Scholar
13. [13]
Y.J. Shen and W. Lin, A wavelet-Gelerkin method for a linear equation system with Hadamard integrals, manuscript. Google Scholar
14. [14]
J.Z. Wang, Cubic spline wavelet bases of Sobolev spaces and multilevel interpolation, Appl. Comput. Harmonic Anal. 3 (1996) 154–163.
15. [15]
J.-C. Xu and W.C. Shann, Galerkin-wavelet methods for two-point boundary value problems, Numer. Math. 63 (1992) 123–144.