Spaces of polynomial and nonpolynomial spline-wavelets

Yu K Dem’Yanovich,A V Lebedeva,T O Evdokimovas,И Г Бурова

doi:10.1051/matecconf/201929204001

Abstract

This paper, discusses spaces of polynomial and nonpolynomial splines suitable for solving the Hermite interpolation problem (with first-order derivatives) and for constructing a wavelet decomposition. Such splines we call Hermitian type splines of the first level. The basis of these splines is obtained from the approximation relations under the condition connected with the minimum of multiplicity of covering every point of (α, β) (almost everywhere) with the support of the basis splines. Thus these splines belong to the class of minimal splines. Here we consider the processing of flows that include a stream of values of the derivative of an approximated function which is very important for good approximation. Also we construct a splash decomposition of the Hermitian type splines on a non-uniform grid.

Highlights

Much attention has been paid to the application of wavelets to solving various problems
The wavelet-Galerkin method is a useful tool for solving differential equations mainly because the conditional number of the stiffness matrix is independent of the matrix size and the number of iterations for solving the discrete problem by the conjugate gradient method is small
In this paper the authors used this basis in the Galerkin method to solve the second-order elliptic problems with Dirichlet boundary conditions in one and two dimensions and by an appropriate post-processing they achieve the LL2-error of order OO(h4), where h is the step size

Summary

Introduction

Much attention has been paid to the application of wavelets to solving various problems. To numerically solve the Burgers’ equation, in this paper [7] the authors propose a general method for constructing wavelet bases on the interval [0,1] derived from symmetric biorthogonal multiwavelets on the real line. They obtain wavelet bases with simple structures on the interval [0,1] from the Hermite cubic splines. In comparison with all other known constructed wavelets on the interval [0,1], authors constructed wavelet bases on the interval [0,1] from the Hermite cubic splines have good approximation and symmetry properties with extremely short supports, and employ a minimum number of boundary wavelets with a very simple structure These desirable properties make them to be of particular interest in numerical algorithms. Sometimes to discuss the situation connected with a segment [a,b] ⊂ (α,β) is difficult.\ we can solve this problem using our method by restricting all functions on the segment

Splines of Hermit Type

Calibration relations for the Hermitian type splines