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 Izv. Akad. Nauk SSSR Ser. Mat., 1983, Volume 47, Issue 5, Pages 1078–1090 (Mi izv1436)

Approximation in the mean of classes of differentiable functions by algebraic polynomials

V. A. Kofanov

Abstract: The exact values $E_n(W^r_L)_L$ are found for the best approximations in the mean of the function classes
$$W^r_L=\{f:f^{(r-1)} is absolutely continuous, \|f^{(r)}\|_L\leqslant1\},\qquad r =2,3,…,$$
by algebraic polynomials of degree at most $n$ on the interval $[-1,1]$. It is proved that $E_n(W^r_L)_L$ coincides with the uniform norm of the perfect spline
$$\frac1{r!}[(x+1)^r+2\sum^{n+1}_{i=1}(-1)^i(x-x_i)^r_+]$$
with nodes $x_i=-\cos\frac{i\pi}{n+2}$.
Bibliography: 6 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1984, 23:2, 353–365

Bibliographic databases:

UDC: 517.5
MSC: Primary 41A10, 41A15, 41A44; Secondary 41A50

Citation: V. A. Kofanov, “Approximation in the mean of classes of differentiable functions by algebraic polynomials”, Izv. Akad. Nauk SSSR Ser. Mat., 47:5 (1983), 1078–1090; Math. USSR-Izv., 23:2 (1984), 353–365

Citation in format AMSBIB
\Bibitem{Kof83} \by V.~A.~Kofanov \paper Approximation in the mean of classes of differentiable functions by algebraic polynomials \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1983 \vol 47 \issue 5 \pages 1078--1090 \mathnet{http://mi.mathnet.ru/izv1436} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=718416} \zmath{https://zbmath.org/?q=an:0564.41003|0551.41013} \transl \jour Math. USSR-Izv. \yr 1984 \vol 23 \issue 2 \pages 353--365 \crossref{https://doi.org/10.1070/IM1984v023n02ABEH001774} 

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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. V. P. Motornyi, O. V. Motornaya, “On the best $L_1$-approximation by algebraic polynomials to truncated powers and to classes of functions with $L_1$-bounded derivative”, Izv. Math., 63:3 (1999), 561–582
2. Semyon Rafalson, “An Extremal Relation of the Theory of Approximation of Functions by Algebraic Polynomials”, Journal of Approximation Theory, 110:2 (2001), 146
3. J. Bustamante, “Best L 1 Approximation of Truncated Functions, Whitney-Type and Borh–Favard-Type Inequalities”, Acta Math. Hungar, 2014
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