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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
Received: 15.03.2019

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}


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