On Five-Diagonal Splitting for Cubic Spline Wavelets with Six Vanishing Moments on a Segment

Abstract

In this study, we use the vanishing property of the first six moments for constructing a splitting algorithm for cubic spline wavelets. First, we construct the corresponding wavelet space that satisfies the orthogonality conditions for all fifth-degree polynomials. Then, using the homogeneous Dirichlet boundary conditions, we adapt spaces to the closed interval. The originality of the study consists in obtaining implicit relations connecting the coefficients of the spline decomposition at the initial scale with the spline coefficients and wavelet coefficients at the nested scale by a tape system of linear algebraic equations with a non-degenerate matrix. After excluding the even rows of the system, in contrast to the case with two zero moments, the resulting transformation matrix has five (instead of three) diagonals. The results of numerical experiments on calculating the derivatives of a discrete function are presented.