Some negative results for the interpolation monotone approximation of functions having a fractional derivative

Abstract

We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function $f ? W^r [0,1]$ by polynomials. The problem of positive approximation is to estimate the pointwise degree of approximation of a function $f ? C^r [0,1] \cap \Delta^0$ where $\Delta^0$ is the set of positive functions on [0,1].

Estimates of the form (1) for positive approximation are known ([1],[2]). The problem of monotone approximation is that of estimating the degree of approximation of a monotone nondecreasing function by monotone nondecreasing polynomials. Estimates of the form (1) for monotone approximation were proved in [3],[4],[8]. In [3],[4] is consider $r ? ?, r > 2$. In [8] is consider $r ? ?, r > 2$. It was proved that for monotone approximation estimates of the form (1) are fails for $r ? ?, r > 2$. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is consider in ([5],[6]). In [5] is consider $r ? ?, r > 2$. In [6] is consider $r ? ?, r > 2$. It was proved that for convex approximation estimates of the form (1) are fails for $r ? ?, r > 2$. In this paper the question of approximation of function $f ? W^r \cap \Delta^1, r ? (3,4)$ by algebraic polynomial $p_n ? \Pi_n \cap \Delta^1$ is consider. The main result of the work generalize the result of work [8] for $r ? (3,4)$.

Key words: approximation of function, Sobolev space, algebraic polynomial, monotone function, convex function.

Pages of the article in the issue: 122 - 127

Language of the article: Ukrainian