Wavelet bases of Hermite cubic splines on the interval

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. 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.