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V. N. Temlyakov

Publications and source records attributed to V. N. Temlyakov.

At least 19 recordsLinked to original sources
arXiv2026

On weak greedy algorithms

The main goal of this paper is twofold. First, we extend some results known in the case of weak greedy algorithms with a scalar parameter to the case of weak greedy algorithms with a weakness sequence. Second, we formulate a new setting of the problem of convergence of greedy algorithms. Usually, we are interested in convergence of an algorithm for all elements of the space. We suggest to study convergence of an algorithm for a subset, which is a generalized octahedron associated with a given dictionary.

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arXiv2025

On stability of Weak Greedy Algorithm in the presence of noise

This paper is devoted to the theoretical study of the efficiency, namely, stability of some greedy algorithms. In the greedy approximation theory researchers are mostly interested in the following two important properties of an algorithm -- convergence and rate of convergence. In this paper we present some results on one more important property of an algorithm -- stability. Stability means that small perturbations do not result in a large change in the outcome of the algorithm. In this paper we discuss one kind of perturbations -- noisy data.

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arXiv2024

Bounds for the sampling discretization error and their applications to the universal sampling discretization

In the first part of the paper we study absolute error of sampling discretization of the integral $L_p$-norm for function classes of continuous functions. We use basic approaches from chaining technique to provide general upper bounds for the error of sampling discretization of the $L_p$-norm on a given function class in terms of entropy numbers in the uniform norm of this class. As an example we apply these general results to obtain new error bounds for sampling discretization of the $L_p$-norms on classes of multivariate functions with mixed smoothness. In the second part of the paper we apply our general bounds to study the problem of universal sampling discretization.

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arXiv2024

Rate of convergence of Thresholding Greedy Algorithms

The rate of convergence of the classical Thresholding Greedy Algorithm with respect to bases is studied in this paper. We bound the error of approximation by the product of both norms -- the norm of $f$ and the $A_1$-norm of $f$. We obtain some results for greedy bases, unconditional bases, and quasi-greedy bases. In particular, we prove that our bounds for the trigonometric basis and for the Haar basis are optimal.

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arXiv2024

Sampling discretization of the uniform norm and applications

Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. Previous known results show that for any $N$-dimensional subspace of the space of continuous functions it is sufficient to use $e^{CN}$ sample points for an accurate upper bound for the uniform norm by the discrete norm and that one cannot improve on the exponential growth of the number of sampling points for a good discretization theorem in the uniform norm. In this paper we focus on two types of results, which allow us to obtain good discretization of the uniform norm with polynomial in $N$ number of points. In the first way we weaken the discretization inequality by allowing a bound of the uniform norm by the discrete norm multiplied by an extra factor, which may depend on $N$. In the second way we impose restrictions on the finite dimensional subspace under consideration. In particular, we prove a general result, which connects the upper bound on the number of sampling points in the discretization theorem for the uniform norm with the best $m$-term bilinear approximation of the Dirichlet kernel associated with the given subspace.

math.FA↗
arXiv2023

On universal sampling recovery in the uniform norm

It is known that results on universal sampling discretization of the square norm are useful in sparse sampling recovery with error measured in the square norm. In this paper we demonstrate how known results on universal sampling discretization of the uniform norm and recent results on universal sampling representation allow us to provide good universal methods of sampling recovery for anisotropic Sobolev and Nikol'skii classes of periodic functions of several variables. The sharpest results are obtained in the case of functions on two variables, where the Fibonacci point sets are used for recovery.

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arXiv2023

On the rate of convergence of greedy algorithms

We prove some results on the rate of convergence of greedy algorithms, which provide expansions. We consider both the case of Hilbert spaces and the more general case of Banach spaces. The new ingredient of the paper is that we bound the error of approximation by the product of both norms -- the norm of $f$ and the $A_1$-norm of $f$. Typically, only the $A_1$-norm of $f$ is used. In particular, we establish that some greedy algorithms (Pure Greedy Algorithm (PGA) and its generalizations) are as good as the Orthogonal Greedy Algorithm (OGA) in this new sense of the rate of convergence, while it is known that the PGA is much worth than the OGA in the standard sense.

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arXiv2022

Sampling discretization error of integral norms for function classes with small smoothness

We consider infinitely dimensional classes of functions and instead of the relative error setting, which was used in previous papers on the integral norm discretization, we consider the absolute error setting. We demonstrate how known results from two areas of research -- supervised learning theory and numerical integration -- can be used in sampling discretization of the square norm on different function classes. We prove a general result, which shows that the sequence of entropy numbers of a function class in the uniform norm dominates, in a certain sense, the sequence of errors of sampling discretization of the square norm of this class. Then we use this result for establishing new error bounds for sampling discretization of the square norm on classes of multivariate functions with mixed smoothness.

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arXiv2022

On universal sampling representation

For the multivariate trigonometric polynomials we study convolution with the corresponding the de la Vallee Poussin kernel from the point of view of discretization. In other words, we replace the normalized Lebesgue measure by a discrete measure in such a way, which preserves the convolution properties and provides sampling discretization of integral norms. We prove that in the two-variate case the Fibonacci point sets provide an ideal (in the sense of order) solution. We also show that the Korobov point sets provide a suboptimal (up to logarithmic factors) solution for an arbitrary number of variables.

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arXiv2020

On the fixed volume discrepancy of the Korobov point sets

This paper is devoted to the study of a discrepancy-type characteristic -- the fixed volume discrepancy -- of the Korobov point sets in the unit cube. It was observed recently that this new characteristic allows us to obtain optimal rate of dispersion from numerical integration results. This observation motivates us to thoroughly study this new version of discrepancy, which seems to be interesting by itself. This paper extends recent results by V. Temlyakov and M. Ullrich on the fixed volume discrepancy of the Fibonacci point sets.

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arXiv2019

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.

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arXiv2017

Remarks on numerical integration, discrepancy, and diaphony

The goal of this paper is twofold. First, we present a unified way of formulating numerical integration problems from both approximation theory and discrepancy theory. Second, we show how techniques, developed in approximation theory, work in proving lower bounds for recently developed new type of discrepancy -- the smooth discrepancy.

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arXiv2017

Fixed volume discrepancy in the periodic case

The smooth fixed volume discrepancy in the periodic case is studied here. It is proved that the Frolov point sets adjusted to the periodic case have optimal in a certain sense order of decay of the smooth periodic discrepancy. The upper bounds for the $r$-smooth fixed volume periodic discrepancy for these sets are established.

math.NA↗
arXiv2017

Dispersion of the Fibonacci and the Frolov point sets

It is proved that the Fibonacci and the Frolov point sets, which are known to be very good for numerical integration, have optimal rate of decay of dispersion with respect to the cardinality of sets. This implies that the Fibonacci and the Frolov point sets provide universal discretization of the uniform norm for natural collections of subspaces of the multivariate trigonometric polynomials. It is shown how the optimal upper bounds for dispersion can be derived from the upper bounds for a new characteristic -- the smooth fixed volume discrepancy. It is proved that the Fibonacci point sets provide the universal discretization of all integral norms.

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arXiv2017

Universal discretization

The paper is devoted to discretization of integral norms of functions from a given collection of finite dimensional subspaces. For natural collections of subspaces of the multivariate trigonometric polynomials we construct sets of points, which are optimally (in the sense of order) good for each subspace of a collection from the point of view of the integral norm discretization. We call such sets universal. Our construction of the universal sets is based on deep results on existence of special nets, known as (t,r,d)-nets.

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arXiv2017

The Marcinkiewicz-type discretization theorems for the hyperbolic cross polynomials

The main goal of this paper is to study the discretization problem for the hyperbolic cross trigonometric polynomials. This important problem turns out to be very difficult. In this paper we begin a systematic study of this problem and demonstrate two different techniques -- the probabilistic and the number theoretical techniques.

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arXiv2017

The Marcinkiewicz-type discretization theorems

The paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. This problem is very important in applications but there is no systematic study of it. We present here a new technique, which works well for discretization of the integral norm. It is a combination of probabilistic technique, based on chaining, with results on the entropy numbers in the uniform norm.

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