Open Access
Access granted
Subscription Access
Vol 216, No 3 (2025)
- Year: 2025
- Articles: 12
- URL: https://journal-vniispk.ru/0368-8666/issue/view/20346
Tribute to Vladimir Mikhailovich Tikhomirov
3-3
From the editors of this issue
4-4
Local controllability and the boundary of the attainable set of a control system
Abstract
5-25
26-48
Kolmogorov widths, Grassmann manifolds and unfoldings of time series
Abstract
Problems in Kolmogorov's theory of widths and the theory of unfoldings of time series are considered. These theories are related by means of the theory of extremal problems on the Grassmann manifolds $G(n,q)$ of $q$-dimensional linear subspaces of $\mathbb R^n$. The necessary information on the manifolds $G(n,q)$ is provided. Using an unfolding of a time series, the concept of the $q$-width of this series is introduced, and the $q$-width of a time series is calculated in the case of the functional of component analysis of the nodes of the unfolding. Using the Schubert basis of a $q$-dimensional linear subspace of $\mathbb R^n$ the concept of time series regression is introduced and its properties are described. An algorithm for the projection of a piecewise linear curve in $\mathbb R^n$ onto the space of unfoldings of time series is described and, on this basis, the concept of an $L$-approximation of a time series is introduced, where $L$ is an arbitrary $q$-dimensional subspace of $\mathbb R^n$. The results of calculations for discretizations of model functions and for time series obtained at a station monitoring the concentration of atmospheric $\mathrm{CO}_2$ are presented. Bibliography: 32 titles.
49-68
Supersmooth tile $\mathrm B$-splines
Abstract
A tile is a self-affine compact subset of $\mathbb R^n$ whose integer translates tile the space. A tile $\mathrm B$-spline is a self-convolution of the characteristic function of the tile, similarly to $\mathrm B$-splines, which are self-convolutions of the characteristic functions of closed intervals. It is known that tile $\mathrm B$-splines, even ones with ‘fractal’ support, can be ‘supersmooth’, that is, their smoothness can exceed that of classical $\mathrm B$-splines of the same order. We evaluate the smoothness of tile $\mathrm B$-splines in $W_2^k(\mathbb R^n)$ by applying a method developed recently and based on Littlewood–Paley type estimates for refinement equations. We adapt this method for tile $\mathrm B$-splines, thereby obtaining 20 families with the property of supersmoothness. We put forward the conjecture, supported by numerical experiments, that this classification is complete if the number of digits is small. Bibliography: 51 titles.
69-95
On the Hamilton–Jacobi theory for nonsmooth variational problems
Abstract
96-107
Sequences of partial sums of multiple trigonometric Fourier series
Abstract
Let $f$ be an integrable $2\pi$-periodic function of $d\ge2$ variables. For a bounded subset $A$ of the $d$-dimensional space let $S_A(f)$ denote the sum of terms of the Fourier series of $f$ with frequencies in $A$. The following problem is addressed: given a sequence $\{A_j\}$ of bounded convex sets, do there exist a function $f$ and a sequence $\{j_\nu\}$ such that $\lim_{\nu\to\infty} |S_{A_{j_\nu}} (f)|=\infty$ almost everywhere? Bibliography: 5 titles.
108-127
Around Strassen's theorems
Abstract
128-155
Autopolar conic bodies and polyhedra
Abstract
An antinorm in a linear space is a concave analogue of a norm. In contrast to norms, antinorms are not defined on the whole space $\mathbb{R}^d$ but on a cone $K\subset \mathbb{R}^d$. They are applied to functional analysis, optimal control and dynamical systems. Level sets of antinorms are called conic bodies and (in the case of piecewise-linear antinorms) conic polyhedra. The basic facts and notions of the ‘concave analysis’ of antinorms such as separation theorems, duality, polars, Minkowski functionals, and so on, are similar to the ones in the standard convex analysis. There are, however, some significant differences. One of them is the existence of many self-dual objects. We prove that there are infinitely many families of autopolar conic bodies and polyhedra in the cone $K=\mathbb{R}^d_+$. For $d=2$ this gives a complete classification of self-dual antinorms, while for $d\ge 3$ there are counterexamples. Bibliography: 29 titles.
156-176
On some Carlson-type inequalities
Abstract
We find the sharp constant in the inequalitywhere $T$ is a cone in $\mathbb R^d$ and the weights $w(\cdot)$, $w_0(\cdot)$ and $\varphi_j(\cdot)$, $j=1,…,d$, are homogeneous measurable functions. We also obtain similar inequalities for some differential operators. Bibliography: 7 titles.
177-190
Optimal recovery of fractional powers of the Laplace difference operator
Abstract
The concept of a fractional power of the Laplace difference operator of a function on an $d$-dimensional lattice is introduced, and the problem of optimal recovery from inaccurate information about the function itself is stated for this fractional power. A family of optimal recovery methods is constructed.Bibliography: 11 titles.
191-202