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

Abstract

The article investigates an implicit method of decomposition of the 7th degree non-Hermitian splines into a series of wavelets with two zero moments. 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 even-odd splitting of the wavelet decomposition algorithm into a solution of the half-size five-diagonal system of linear equations and some local averaging formulas are substantiated. The results of numerical experiments on accuracy on polynomials and compression of spline-wavelet decomposition are presented.

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

“…Recall that for the case of interpolation by splines on a finite interval [0, 2 L ], the most productive approach to constructing basis functions is to set multiple nodes at the ends of the interval, which corresponds to zeroing of the approximating spline and some of its derivatives at the ends of the interval [2]. Then the left seventh degree basic functions have the view forms [18]. They have the following supports,…”

“…As to boundary seventh-degree basic wavelets, then because of the future need for splitting the decomposition matrix we can use the method of constructing wavelets that are orthogonal to all firstdegree polynomials and include only even basic splines [18],…”

“…The graphs of basis spline functions and wavelets of the 7th degree, orthogonal to all polynomials of the 1st degree, were shown in [18].…”

“…The first attempt to find out whether there is an algorithm for splitting non-orthogonal wavelets of the seventh degree was made in the work of the author [18], the method of five-diagonal splitting was studied, and the absence of strict diagonal dominance in the resulting system was established. In this article, we justify the stability of the solution for the sevendiagonal splitting method; presents the results of numerical experiments on the approximation of a discretely given function.…”