Representation of the approximational error term and sharp estimates for some local splines

Abstract

In the present paper we study local splines of defect 1 on a uniform net with step approximating functions and their derivatives of arbitrary order. Namely, splines of the first, third, and fifth degrees realizing different orders of approximation of f(s) are considered, including splines realizing the maximally possible orders O(~2), O(~), O(~6), respectively, for the minimal number of net values of the function f operated on (so-called splines of minimal model, SMM). In addition, the simplest splines of arbitrary degree exactly reproducing the derivatives ps+1(s)(t) of polynomials of degree s + 1 are considered, and also SMM of arbitrary degree r exactly reproducing Pr(t). In the paper we find explicit expressions for the error terms arising in the approximation by the splines listed above of functions and their derivatives of arbitrary order. The expressions obtained let us find sharp estimates for the error of approximation on the corresponding classes of functions. These results are based on the asymptotic expansions of the error terms in powers of ~ established in [I, 2] and also on one of the results of [3].