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

A. A. Vasil'eva

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 2, pp. 48-66

Résumé

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 $y_1, \, \dots, \, y_n$. 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: $M=\{ f\in AC[0,\,1]:\;\|\dot{f}\|_{L_p[0, \, 1]}\le 1,\; \|\dot{f}\|_{L_1\left[ (j-1)/s, \, j/s\right]} \le \varepsilon_j/s, \; 1\le j \le s\}$. 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 $\varphi(\varepsilon_1, \, \dots, \, \varepsilon_s)n^{-1}$, where $\varphi(\varepsilon_1, \, \dots, \, \varepsilon_s) \to 0$ as $(\varepsilon_1, \, \dots, \, \varepsilon_s) \to 0$ (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.

Détail

Keywords: Kolmogorov widths, Sobolev classes
Mots-clés : interpolation classes.