"Узагальнення поточкових інтерполяційних оцінок опуклого наближення функцій, що мають дробову похідну "

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 problem of positive approximation is to estimate the pointwise degree of approximation of a function of r times continuously differentiable and 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 is natural and r not equal one. In [8] is consider r is real and r more 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 approximation of a convex function by convex polynomials. The problem of convex approximation is consider in ([5, 6]). In [5] is consider r is natural and r not equal one. In [6] is consider r is real and r more two. It was proved that for convex approximation estimates of the form (1) are fails for r is real and r more two. In [9] the question of approximation of function of Sobolev`s space and convex by algebraic convex polynomial is consider. It is proved, that for this function, estimate (1) is not true, if r is more three and less four generally speaking. In this paper the question of approximation of function Sobolev`s space and convex by algebraic convex polynomial is consider. This paper is the generalization of results papers [9] and [11]. It is proved, that for function of Sobolev`s space and convex, estimate of the type (1) is not true, generally speaking. The main result is the ana­log of the theorem 2.3 in [11].