arXiv subjects

S. Tikhonov

Publications and source records attributed to S. Tikhonov.

9 recordsLinked to original sources

arXiv2022

On the cardinality of lower sets and universal discretization

A set $Q$ in $\mathbb{Z}_+^d$ is a lower set if $(k_1,\dots,k_d)\in Q$ implies $(l_1,\dots,l_d)\in Q$ whenever $0\le l_i\le k_i$ for all $i$. We derive new and refine known results regarding the cardinality of the lower sets of size $n$ in $\mathbb{Z}_+^d$. Next we apply these results for universal discretization of the $L_2$-norm of elements from $n$-dimensional subspaces of trigonometric polynomials generated by lower sets.

math.NA↗

arXiv2020

Smoothness of functions vs. smoothness of approximation processes

We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed: the first based on geometric properties of Banach spaces and the second on Littlewood-Paley and Hörmander type multiplier theorems. In particular, we obtain new sharp inequalities for measures of smoothness given by the $K$-functionals or moduli of smoothness. As examples of approximation processes we consider best polynomial and spline approximations, Fourier multiplier operators on $\mathbb{T}^d$, $\mathbb{R}^d$, $[-1, 1]$, nonlinear wavelet approximation, etc.

math.CA↗

arXiv2020

Entropy numbers and Marcinkiewicz-type discretization theorem

This paper studies the behavior of the entropy numbers of classes of functions with bounded integral norms from a given finite dimensional linear subspace. Upper bounds of these entropy numbers in the uniform norm are obtained and applied to establish a Marcinkiewicz type discretization theorem for integral norms of functions from a given finite dimensional subspace.

math.CA↗

arXiv2020

Sampling discretization of integral norms

The paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. Even though this problem is extremely important in applications, its systematic study has begun recently. In this paper we obtain a conditional theorem for all integral norms $L_q$, $1\le q<\infty$, which is an extension of known results for $q=1$. To discretize the integral norms successfully, we introduce a new technique, which is a combination of probabilistic technique with results on the entropy numbers in the uniform norm. As an application of the general conditional theorem, we derive a new Marcinkiewicz type discretization for the multivariate trigonometric polynomials with frequencies from the hyperbolic crosses.

math.CA↗

arXiv2018

Sharp Remez inequality

Let an algebraic polynomial $P_n(ζ)$ of degree $n$ be such that $|P_n(ζ)|\le 1$ for $ζ\in E\subset\mathbb{T}$ and $|E|\ge 2π-s$. We prove the sharp Remez inequality $$ \sup_{ζ\in\mathbb{T}}|P_n(ζ)|\le \mathfrak{T}_{n}\left(\sec \frac{s} 4\right),$$ where $\mathfrak{T}_{n}$ is the Chebyshev polynomial of degree $n$. The equality holds if and only if $$ P_n(e^{iz})=e^{i(nz/2+c_1)}\mathfrak{T}_n\left(\sec\frac s 4\cos \frac {z-c_0} 2\right), \quad c_0,c_1\in\mathbb{R}. $$ This gives the solution of the long-standing problem on the sharp constant in the Remez inequality for trigonometric polynomials.

math.CA↗

arXiv2018

Integral norm discretization and related problems

The problem of replacing an integral norm with respect to a given probability measure by the corresponding integral norm with respect to a discrete measure is discussed in the paper. The above problem is studied for elements of finite dimensional spaces. Also, discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. We pay special attention to the case of the multivariate trigonometric polynomials with frequencies from a finite set with fixed cardinality. Both new results and a survey of known results are presented.

math.NA↗

arXiv2016

Remez-type inequalities for the hyperbolic cross polynomials

In this paper we study the Remez-type inequalities for trigonometric polynomials with harmonics from hyperbolic crosses. The interrelation between the Remez and Nikolskii inequalities for individual functions and its applications are discussed.

math.CA↗

arXiv2012

Moduli of smoothness and growth properties of Fourier transforms: two-sided estimates

We prove two-sided inequalities between the integral moduli of smoothness of a function on $\mathbb{R}^d/\mathbb{T}^d$ and the weighted tail-type integrals of its Fourier transform/series. Sharpness of obtained results in particular is given by the equivalence results for functions satisfying certain regular conditions. Applications include a quantitative form of the Riemann-Lebesgue lemma as well as several other questions in approximation theory and the theory of function spaces.

math.CA↗

arXiv2012

Embedding theorems of function classes

In this paper we obtain the necessary and sufficient conditions for embedding results of different function classes. The main result is a criterion for embedding theorems for the so-called generalized Weyl-Nikol'skii class and the generalized Lipschitz class. To define the Weyl-Nikol'skii class, we use the concept of a $(λ,β)$-derivative, which is a generalization of the derivative in the sense of Weyl. As corollaries, we give estimates of norms and moduli of smoothness of transformed Fourier series.

math.CA↗