Vladimir Tikhomirov

ε-Entropy and ε-Capacity of Sets In Functional Spaces

The article is mainly devoted to the systematic exposition of results that were published in the years 1954–1958 by K. I. Babenko [1], A. G. Vitushkin [2,3], V. D. Yerokhin [4], A. N. Kolmogorov [5,6] and V. M. Tikhomirov [7]. It is natural that when these materials were systematically rewritten, several new theorems were proved and certain examples were computed in more detail. This review also incorporates results not published previously which go beyond the framework of such a systematization, and belong to V. I. Arnold (§6) and V. M. Tikhomirov (§§4,7 and §8).

Wavelet compression and nonlinearn-widths

It is shown that certain algorithms of compression based on wavelet decompositions are optimal in the sense of nonlinearn-widths.

ε -Entropy and ε -Capacity

The topic of the given commentary is directly related to the papers [1–5]. However, it is natural to consider a somewhat wider cycle in which we shall include [6–10]. The backbone bringing together the entire cycle of articles [1–10] is the notion of entropy.

Optimal Recovery and Extremum Theory

In this paper optimal recovery problems of linear functionals on classes of smooth and analytic functions on the basis of linear information are considered from the general viewpoint of extremum theory. A general result about the connection of optimal recovery method with Lagrange multipliers of some convex extremal problem is applied to the analysis of classical recovery problems on the generalized Sobolev, Hardy, and Hardy-Sobolev classes.

Newton’s method, differential equations, and the lagrangian principle for necessary extremum conditions

We show how one can use a modified Newton’s method to prove existence and uniqueness theorems for solutions of differential equations and theorems on the continuous and differentiable dependence of these solutions on the initial data and parameters and to derive necessary conditions for an extremum in various extremum problems (from the origins to our days).

Indefinite Knowledge about an Object and Accuracy of Its Recovery Methods

An approach to the problem of optimal recovery of functionals and operators on classes of functions under the conditions of infinite knowledge of functions themselves is discussed. The capabilities of this approach are demonstrated in a number of examples. In the end of the paper, a general result about optimal recovery of linear functionals is given.

On Operations on Sets

In 1921 N.N. Luzin delivered a series of lectures on the theory of functions at Moscow University, in which he defined the class of C-sets and posed the problem of comprehensive study of such sets. Recently E.A. Selivanovskii1 has published several results relating to this class and has given it a rather complete characterization. In this connection, the investigation presented below would seem to be of interest, since I obtain some of these results proceeding from very general considerations. The class of C-sets, as well as the class of B-measurable sets splits into non-empty K1 classes, and it turns out that the same is true for an arbitrary class of sets generated by iteration of operations of a very general type, defined below, and the operation of taking a complement.

Average dimension and v -widths of classes of functions on the whole line

The first works devoted to the problem of approximation of functions on R were written by Serge Bernstein in the thirties. He considered the space of entire functions of exponential type as a tool of approximation. But during the last decades splines have been used more and more often as an approximating set. Spaces of entire functions and splines are of infinite dimension and quantities, characterizing the corresponding approximation, are expressed in terms reflecting the inner structure of the approximating set (such as order of the entire function or density of distribution of spline knots). The following question arises: how can we compare these methods of approximation? We represent an approach to this problem which is connected with the concept of average dimension. C. Shannon was the first to consider the problem of average characteristics, which was connected with the notion of “average entropy.” A. Kolmogorov introduced the nonrandom version of this concept and the first result in that direction was obtained by Tikhomirov (1957). An analogous characteristic, namely average dimension, which is based on the notion of Kolmogorov’s n-width, was proposed by Tikhomirov (1980) and the program of exploration was outlined (see also Tikhomirov, 1983). The works of Tikhomirov’s students (DinhDung and Magaril-Il’yaev, 1979; Le Chong Tung, 1980; Dinh-Dung, 1980) were devoted to the calculation of average and a-average dimensions of some classes of functions.

In memory of Nikolai Sergeevich Bakhvalov (1934-2005)

Nikolai Sergeevich Bakhvalov was an outstanding mathematician, an Academician of the Russian Academy Sciences, a Professor of the Moscow State University (msu), and the chair of the Computational Mathematics Section in the Mechanics and Mathematics Department at msu. We have split our personal recollections of Bakhvalov into two parts. In Part?I (which was written by the first author), we present the recollections of peers; in Part?II (written by the second author), we present recollections of generations of Bakhvalovs students.

On the Pasture Territories Covering Maximal Grass

In this chapter we consider the so-called pasture territory problem, its basic elements, and some related extremal problems. We describe the pasture territory as a graph of a piecewise smooth and continuous function f(x, y) defined on a closed, connected domain of a plane. Considering extremal problems is related with finding the location of the nomadic residence, when the exploiting pasture territory has maximum grass mass, and finding the bound of the territory, when the place of the residence is fixed [1, 2, 5].