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 Mat. Zametki, 2010, Volume 87, Issue 5, Pages 669–683 (Mi mz8716)

On the Exact Values of the Best Approximations of Classes of Differentiable Periodic Functions by Splines

V. F. Babenkoab, N. V. Parfinovichb

a Institute of Applied Mathematics and Mechanics, Ukraine National Academy of Sciences
b Dnepropetrovsk National University

Abstract: We obtain the exact values of the best $L_1$-approximations of classes $W^rF$, $r\in\mathbb N$, of periodic functions whose $r$th derivative belongs to a given rearrangement-invariant set $F$, as well as of classes $W^rH^\omega$ of periodic functions whose $r$th derivative has a given convex (upward) majorant $\omega(t)$ of the modulus of continuity, by subspaces of polynomial splines of order $m\ge r+1$ and of deficiency 1 with nodes at the points $2k\pi/n$ and $2k\pi/n+h$, $n\in\mathbb N$, $k\in\mathbb Z$, $h\in(0,2\pi/n)$. It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding functional classes.

Keywords: best approximation, differentiable periodic function, polynomial spline, Kolmogorov width, modulus of continuity, extremal subspace, Jackson-type inequality, the space $L_1$, Sobolev class $W_p^r$, the space $L_p$, Orlicz space

DOI: https://doi.org/10.4213/mzm8716

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English version:
Mathematical Notes, 2010, 87:5, 623–635

Bibliographic databases:

UDC: 517

Citation: V. F. Babenko, N. V. Parfinovich, “On the Exact Values of the Best Approximations of Classes of Differentiable Periodic Functions by Splines”, Mat. Zametki, 87:5 (2010), 669–683; Math. Notes, 87:5 (2010), 623–635

Citation in format AMSBIB
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\paper On the Exact Values of the Best Approximations of Classes of Differentiable Periodic Functions by Splines
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\pages 669--683
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\crossref{https://doi.org/10.4213/mzm8716}
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\transl
\jour Math. Notes
\yr 2010
\vol 87
\issue 5
\pages 623--635
\crossref{https://doi.org/10.1134/S0001434610050032}
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• http://mi.mathnet.ru/eng/mz8716
• https://doi.org/10.4213/mzm8716
• http://mi.mathnet.ru/eng/mz/v87/i5/p669

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This publication is cited in the following articles:
1. Vasil'eva A.A., “Widths of Weighted Sobolev Classes With Constraints F(a) = Center Dot Center Dot Center Dot = F(K-1)(a) = F(K)(B) = Center Dot Center Dot Center Dot = F(R-1)(B)=0 and the Spectra of Nonlinear Differential Equations”, Russ. J. Math. Phys., 24:3 (2017), 376–398
2. Parfinovych N.V., “Exact Values of the Best (Oe > 1/4, Beta)-Approximations For the Classes of Convolutions With Kernels That Do Not Increase the Number of Sign Changes”, Ukr. Math. J., 69:8 (2018), 1248–1261
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