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 Trudy Inst. Mat. i Mekh. UrO RAN, 2019, Volume 25, Number 2, Pages 48–66 (Mi timm1623)

Kolmogorov widths of Sobolev classes on a closed interval with constraints on the variation

A. A. Vasil'eva

Lomonosov Moscow State University

Abstract: We study the problem of estimating Kolmogorov widths in $L_q[0, 1]$ for the Lipschitz classes of functions with fixed values at several points: $\tilde M=\{f\in AC[0, 1],\; \|\dot{f}\|_\infty \le 1, \; f(j/s)=y_j, \; 0\le j\le s\}$. Applying well-known results about the widths of Sobolev classes, it is easy to obtain order estimates up to constants depending on $q$ and $y_1, …, y_n$. Here we obtain order estimates up to constants depending only on $q$. To this end, we estimate the widths of the intersection of two finite-dimensional sets: a cube and a weighted Cartesian product of octahedra. If we take the unit ball of $l_p^n$ instead of the cube, we get a discretization of the problem on estimating the widths of the intersection of the Sobolev class and the class of functions with constraints on their variation: $M=\{ f\in AC[0, 1]:\;\|\dot{f}\|_{L_p[0, 1]}\le 1,\; \|\dot{f}\|_{L_1[ (j-1)/s, j/s]} \le \varepsilon_j/s, \; 1\le j \le s\}$. For sufficiently large $n$, order estimates are obtained for the widths of these classes up to constants depending only on $p$ and $q$. If $p>q$ or $p>2$, then these estimates have the form $\varphi(\varepsilon_1, …, \varepsilon_s)n^{-1}$, where $\varphi(\varepsilon_1, …, \varepsilon_s) \to 0$ as $(\varepsilon_1, …, \varepsilon_s) \to 0$ (explicit formulas for $\varphi$ are given in the paper). If $p\le q$ and $p\le 2$, then the estimates have the form $n^{-1}$ (hence, the constraints on the variation do not improve the estimate for the widths). The upper estimates are proved with the use of Galeev's result on the intersection of finite-dimensional balls, whereas the proof of the lower estimates is based on a generalization of Gluskin's result on the width of the intersection of a cube and an octahedron.

Keywords: Kolmogorov widths, Sobolev classes, interpolation classes.

 Funding Agency Grant Number Russian Foundation for Basic Research 19-01-00332 This work was supported by the Russian Foundation for Basic Research (project no. 19-01-00332).

DOI: https://doi.org/10.21538/0134-4889-2019-25-2-48-66

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UDC: 517.518.224
MSC: 41A46

Citation: A. A. Vasil'eva, “Kolmogorov widths of Sobolev classes on a closed interval with constraints on the variation”, Trudy Inst. Mat. i Mekh. UrO RAN, 25, no. 2, 2019, 48–66

Citation in format AMSBIB
\Bibitem{Vas19} \by A.~A.~Vasil'eva \paper Kolmogorov widths of Sobolev classes on a closed interval with constraints on the variation \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2019 \vol 25 \issue 2 \pages 48--66 \mathnet{http://mi.mathnet.ru/timm1623} \crossref{https://doi.org/10.21538/0134-4889-2019-25-2-48-66} \elib{http://elibrary.ru/item.asp?id=38071599}