arXiv subjects

V. Temlyakov

Publications and source records attributed to V. Temlyakov.

28 records · Page 2Linked to original sources

arXiv2021

Sampling discretization of integral norms and its application

The paper addresses the problem of sampling discretization of integral norms of elements of finite-dimensional subspaces satisfying some conditions. We prove sampling discretization results under two standard kinds of assumptions -- conditions on the entropy numbers and conditions in terms of the Nikol'skii-type inequalities. We prove some upper bounds on the number of sample points sufficient for good discretization and show that these upper bounds are sharp in a certain sense. Then we apply our general conditional results to subspaces with special structures, namely, subspaces with the tensor product structure. We demonstrate that applications of results based on the Nikol'skii-type inequalities provide somewhat better results than applications of results based on the entropy numbers conditions. Finally, we apply discretization results to the problem of sampling recovery.

math.NA↗

arXiv2021

Universal sampling discretization

Let $X_N$ be an $N$-dimensional subspace of $L_2$ functions on a probability space $(Ω, μ)$ spanned by a uniformly bounded Riesz basis $Φ_N$. Given an integer $1\leq v\leq N$ and an exponent $1\leq q\leq 2$, we obtain universal discretization for integral norms $L_q(Ω,μ)$ of functions from the collection of all subspaces of $X_N$ spanned by $v$ elements of $Φ_N$ with the number $m$ of required points satisfying $m\ll v(\log N)^2(\log v)^2$. This last bound on $m$ is much better than previously known bounds which are quadratic in $v$. Our proof uses a conditional theorem on universal sampling discretization, and an inequality of entropy numbers in terms of greedy approximation with respect to dictionaries.

math.FA↗

arXiv2020

Bounds on Kolmogorov widths and sampling recovery for classes with small mixed smoothness

Results on asymptotic characteristics of classes of functions with mixed smoothness are obtained in the paper. Our main interest is in estimating the Kolmogorov widths of classes with small mixed smoothness. We prove the corresponding bounds for the unit balls of the trigonometric polynomials with frequencies from a hyperbolic cross. We demonstrate how our results on the Kolmogorov widths imply new upper bounds for the optimal sampling recovery in the $L_2$ norm of functions with small mixed smoothness.

math.NA↗

arXiv2020

On optimal recovery in $L_2$

We prove that the optimal error of recovery in the $L_2$ norm of functions from a class $\bF$ can be bounded above by the value of the Kolmogorov width of $\bF$ in the uniform norm. We demonstrate on a number of examples of $\bF$ from classes of functions with mixed smoothness that the obtained inequality provides a powerful tool for estimating errors of optimal recovery.

math.NA↗

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↗

arXiv2019

Sparse approximation of individual functions

Results on two different settings of asymptotic behavior of approximation characteristics of individual functions are presented. First, we discuss the following classical question for sparse approximation. Is it true that for any individual function from a given function class its sequence of errors of best sparse approximations with respect to a given dictionary decays faster than the corresponding supremum over the function class? Second, we discuss sparse approximation by greedy type algorithms. We show that for any individual function from a given class we can improve the upper bound on the rate of convergence of the error of approximation by a greedy algorithm if we use some information from the previous iterations of the algorithm. We call bounds of this type a posteriori bounds.

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↗

arXiv2016

On the entropy numbers of the mixed smoothness function classes

Behavior of the entropy numbers of classes of multivariate functions with mixed smoothness is studied here. This problem has a long history and some fundamental problems in the area are still open. The main goal of this paper is to develop a new method of proving the upper bounds for the entropy numbers. This method is based on recent developments of nonlinear approximation, in particular, on greedy approximation. This method consists of the following two steps strategy. At the first step we obtain bounds of the best m-term approximations with respect to a dictionary. At the second step we use general inequalities relating the entropy numbers to the best m-term approximations. For the lower bounds we use the volume estimates method, which is a well known powerful method for proving the lower bounds for the entropy numbers. It was used in a number of previous papers.

math.NA↗

arXiv2014

Nonlinear tensor product approximation of functions

We are interested in approximation of a multivariate function $f(x_1,\dots,x_d)$ by linear combinations of products $u^1(x_1)\cdots u^d(x_d)$ of univariate functions $u^i(x_i)$, $i=1,\dots,d$. In the case $d=2$ it is a classical problem of bilinear approximation. In the case of approximation in the $L_2$ space the bilinear approximation problem is closely related to the problem of singular value decomposition (also called Schmidt expansion) of the corresponding integral operator with the kernel $f(x_1,x_2)$. There are known results on the rate of decay of errors of best bilinear approximation in $L_p$ under different smoothness assumptions on $f$. The problem of multilinear approximation (nonlinear tensor product approximation) in the case $d\ge 3$ is more difficult and much less studied than the bilinear approximation problem. We will present results on best multilinear approximation in $L_p$ under mixed smoothness assumption on $f$.

stat.ML↗