Узагальнення негативних результатів для інтерполяційного опуклого наближення функцій, що мають дробову похідну в просторі Соболєва

Abstract

We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function of Sobolev`s space by polynomials. The prob-lem of positive approximation is to estimate the pointwise degree of approxi-mation of a function of r times continuously differentiable and positive func-tions 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 de-gree of approximation of a monotone nondecreasing function bymonotone nondecreasing polynomials. Estimates of the form (1) for monotone approxi-mation were proved in [3, 4, 7]. In [3, 4] is consider r is natural and r not equal one. In [7] is consider r is real and rmore two. It was proved that for monotone approximation estimates of the form (1) are fails for r is real and r more two. The problem of convex approximation is that of estimating the degree of ap-proximation of a convex function by convex polynomials. The problem of con-vex approximation is consider in [8-10]. In [8] is consider r is natural and r not equal one. In [9] is consider r is real and r more two. It was proved that for con-vex approximation estimates of the form (1) are fails for r is real and r more two. In [10] the question of approximation of function of Sobolev`s space and convex by algebraic convex polynomial is consider, if the index of the Sobolev space is in the interval from three to four. It is proved that the estimate that gen-eralizes (1) is false This paper investigates the issue of approximation of con-vex functions from the Sobolev space by convex algebraic polynomials for a real index of the Sobolev space from the interval from two to three.Similarly to the paper [10], a counterexample is built, which shows that the estimate that generalizes the estimate (1) is false.This paper isthe generalization of results papers [9] and [11]. The main result is the analog of the theorem 2.3 in [11].