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 Mat. Zametki, 1971, Volume 9, Issue 4, Pages 441–447 (Mi mz7028)

Algebraic-polynomial approximation of functions satisfying a Lipschitz condition

N. P. Korneichuka, A. I. Polovinab

a Dnepropetrovsk State University
b Kommunarskii Mining and Metallurgical Institute

Abstract: For functions $f(x)\in KH^{(\alpha)}$ (satisfying the Lipschitz condition of order $\alpha$ ($0<\alpha<1$) with constant $K$ on $[-1, 1]$), the existence is proved of a sequence $P_n(f; x)$ of algebraic polynomials of degree $n=1, 2, …$, such that $|f(x)-P_{n-1}(f; x)|\leqslant\sup\limits_{f\in KH^{(\alpha)}}E_n(f)[(1-x^2)^{\alpha/2}+o(1)]$ when $n\to\infty$, uniformly for $x\in[-1, 1]$ , where $E_n(f)$ is the best approximation of $f(x)$ by polynomials of degree not higher than $n$.

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English version:
Mathematical Notes, 1971, 9:4, 254–257

Bibliographic databases:

UDC: 517.5

Citation: N. P. Korneichuk, A. I. Polovina, “Algebraic-polynomial approximation of functions satisfying a Lipschitz condition”, Mat. Zametki, 9:4 (1971), 441–447; Math. Notes, 9:4 (1971), 254–257

Citation in format AMSBIB
\Bibitem{KorPol71} \by N.~P.~Korneichuk, A.~I.~Polovina \paper Algebraic-polynomial approximation of functions satisfying a~Lipschitz condition \jour Mat. Zametki \yr 1971 \vol 9 \issue 4 \pages 441--447 \mathnet{http://mi.mathnet.ru/mz7028} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=287238} \zmath{https://zbmath.org/?q=an:0226.41002|0216.38902} \transl \jour Math. Notes \yr 1971 \vol 9 \issue 4 \pages 254--257 \crossref{https://doi.org/10.1007/BF01387776} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. N. P. Korneichuk, “S. M. Nikol'skii and the development of research on approximation theory in the USSR”, Russian Math. Surveys, 40:5 (1985), 83–156
2. V. P. Motornyi, “Approximation of a Class of Singular Integrals by Algebraic Polynomials with Regard to the Location of a Point on an Interval”, Proc. Steklov Inst. Math., 232 (2001), 260–277
3. A. V. Mironenko, “On the Jackson–Stechkin inequality for algebraic polynomials”, Proc. Steklov Inst. Math. (Suppl.), 273, suppl. 1 (2011), S116–S123
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