Affiliation:
1. Taras Shevchenko National University of Kyiv
Abstract
We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function f \in W^r [0,1] by polynomials. The problem of positive approximation is to estimate the pointwise degree of approximation of a function f \in C^r [0,1] \Wedge \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 \in N, r>2. In [8] is consider r \in R, r>2. It was proved that for monotone approximation estimates of the form (1) are fails for r \in 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],[11]). In [5] is consider r \in N, r>2. It was proved that for convex approximation estimates of the form (1) are fails for r \in N, r>2. In [6] is consider r \in R, r\in(2;3). It was proved that for convex approximation estimates of the form (1) are fails for r \in R, r\in(2;3). In [11] is consider r \in R, r\in(3;4). It was proved that for convex approximation estimates of the form (1) are fails for r \in R, r\in(3;4). In [9] is consider r \in R, r>4. It was proved that for f \in W^r [0,1] \Wedge \Delta^2, r>4 estimate (1) is not true. In this paper the question of approximation of function f \in W^r [0,1] \Wedge \Delta^2, r>4 by algebraic polynomial p_n \in \Pi_n \Wedge \Delta^2 is consider. It is proved, that for f \in W^r [0,1] \Wedge \Delta^2, r>4, estimate (1) can be improved, generally speaking.
Publisher
Taras Shevchenko National University of Kyiv
Subject
Medical Assisting and Transcription,Medical Terminology
Reference11 articles.
1. TELYAKOVSKIJ S.A. (1966) Dve teoremy o priblizhenii funkcij algebraicheskimi polinomami, Mat. sb. 79, p. 252-265
2. GOPENGAUZ A.I. (1994) Pointwise estimates of Hermitian interpolation 77, p. 34-43
3. DeVore R.A. and YU X.M. (1985) Pointwise estimates for monotone polynomial approximation, Constr. Approx.,1 p. 323–331.
4. H. H. GONSKA, D. LEVIATAN, I. A. SHEVCUK, and H. -J. WENZ (2000) Interpolatory pointwise estimates for polynomial approximations, Constr. Approx. 16, p. 603–629.
5. PETROVA T.O., (2005) Kontrpryklad u interpolyatsiinomu opuklomu nablyzhenni, Pratsi Instytutu matematyky NAN Ukrainy “Matematyka ta zastosuvannia. Teoriya nablyzhennia functsii”, 35 p. 107–112.