On the seven-diagonals splitting for the cubic spline wavelets with six vanishing moments on an interval

Abstract

This study uses a zeroing property of the first six moments for constructing a splitting algorithm for the cubic spline wavelets. First, we construct a system of cubic basic spline-wavelets, realizing orthogonal conditions to all polynomials up to any degree. Then, using the homogeneous Dirichlet boundary conditions, we adapt spaces to the orthogonality to all polynomials up to the fifth degree on the closed interval. The originality of the study consists of obtaining implicit finite 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, the resulting transformation matrix has seven diagonals, instead of five as in the previous case with four zero moments. A modification of the system is performed, which ensures a strict diagonal dominance, and, consequently, the stability of the calculations. The comparative results of numerical experiments on approximating and calculating the derivatives of a discrete function are presented.

An Algorithm with the Even-odd Splitting of the Wavelet Transform of Non-Hermitian Splines of the Seventh Degree, II

“…Now we will use the method of even-odd splitting to the resulting system solving [17]. We choose for this purpose some preconditioning matrix R L to receive an easy invertible matrix…”

“…Meanwhile, in the work of the author [17], nonorthogonal wavelets of the third degree with the first six zero moments, i.e., orthogonal to all polynomials of the fifth degree, were considered; the existence of finite implicit decomposition relations was proved and an efficient even-odd splitting algorithm based on them for wavelet analysis was substantiated. The importance of the new algorithm in wavelet theory lies in its stability and ease of implementation, since at each resolution step a matrix with strict diagonal dominance is solved.…”