Construction of an Effective Preconditioner for the Even-odd Splitting of Cubic Spline Wavelets

In this study, the method for decomposing splines of degree m and smoothness C^m-1 into a series of wavelets with zero moments is investigated. The system of linear algebraic equations connecting the coefficients of the spline expansion on the initial scale with the spline coefficients and wavelet coefficients on the embedded scale is obtained. The originality consists in the application of some preconditioner that reduces the system to a simpler band system of equations. Examples of applying the method to the cases of first-degree spline wavelets with two first zero moments and cubic spline wavelets with six first zero moments are presented. For the cubic case after splitting the system into even and odd rows, the resulting matrix acquires a seven-diagonals form with strict diagonal dominance, which makes it possible to apply an effective sweep method to its solution

Shifted Cubic Spline Wavelets with Two Vanishing Moments on the Interval and a Splitting Algorithm

This paper deals with the use of the first two vanishing moments for constructing cubic spline-wavelets orthogonal to polynomials of the first degree. A decrease in the supports of these wavelets is shown in comparison with the classical semiorthogonal wavelets. For splines with homogeneous Dirichlet boundary conditions of the second order, an algorithm of the shifted wavelet transform is obtained in the form of a solution of a tridiagonal system of linear equations with a strict diagonal dominance