One counterexample for convex approximation of function with fractional derivatives, r>4
Тип публікації :
Стаття
Дата випуску :
2018
Автор(и) :
Petrova, T. O.
Мова основного тексту :
Англійська
eKNUTSHIR URL :
Випуск :
3
ISSN :
1812-5409
Початкова сторінка :
53
Кінцева сторінка :
56
Цитування :
[APA 7] Petrova, T. O. (2018). One counterexample for convex approximation of function with fractional derivatives, r>4. Bulletin of Taras Shevchenko National University of Kyiv. Physics and Mathematics, (3), 53–56. https://doi.org/10.17721/1812-5409.2018/3.7
[ДСТУ] Petrova T. O. One counterexample for convex approximation of function with fractional derivatives, r>4. Bulletin of Taras Shevchenko National University of Kyiv. Physics and Mathematics. 2018. no. 3. P. 53—56. DOI: 10.17721/1812-5409.2018/3.7 (date of access: 25.07.2026).
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 ofapproximation 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.Key words: approximation of function, Sobolev space, algebraic polynomial, monotone function, convex function.Pages of the article in the issue: 53 - 56Language of the article: Ukrainian
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