Sampling Discretization of Integral Norms of the Hyperbolic Cross Polynomials (original) (raw)

The Marcinkiewicz-type discretization theorems for the hyperbolic cross polynomials

arXiv (Cornell University), 2017

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 techniquesthe probabilistic and the number theoretical techniques.

Sampling discretization of integral norms

2020

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< q<∞, 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.

Sampling discretization of the uniform norm

2021

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 trigonometric polynomials with frequencies from an arbitrary finite set with fixed cardinality. We give two different proofs of the fact that for any N -dimensional subspace of the space of continuous functions it is sufficient to use eCN sample points for an accurate upper bound for the uniform norm. Previous known results show that one cannot improve on the exponential growth of the number of sampling points for a good discretization theorem in the uniform norm. Also, 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. We illustrate application of our technique on the example of trigonometric polynomials.

A remark on discretization of the uniform norm

arXiv (Cornell University), 2021

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 trigonometric polynomials with frequencies from an arbitrary finite set with fixed cardinality. We give two different proofs of the fact 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. Previous known results show that one cannot improve on the exponential growth of the number of sampling points for a good discretization theorem in the uniform norm. Also, 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. We illustrate the application of our technique on the example of trigonometric polynomials.

Integral norm discretization and related problems

Russian Mathematical Surveys, 2019

The problem is discussed of replacing an integral norm with respect to a given probability measure by the corresponding integral norm with respect to a discrete measure. This problem is investigated for elements of finite-dimensional spaces. Also, discretization of the uniform norm of functions in a given finite-dimensional subspace of continuous functions is studied. Special attention is given to the case of multivariate trigonometric polynomials with frequencies (harmonics) in a finite set with fixed cardinality. Both new results and a survey of known results are presented. Bibliography: 47 titles.

Some improved bounds in sampling discretization of integral norms

Journal of Functional Analysis

The paper addresses a problem of sampling discretization of integral norms of elements of finite-dimensional subspaces satisfying some conditions. We prove sampling discretization results under a standard assumption formulated in terms of the Nikol'skii-type inequality. In particular, we obtain some upper bounds on the number of sample points sufficient for good discretization of the integral L p norms, 1 ≤ p < 2, of functions from finite-dimensional subspaces of continuous functions. Our new results improve upon the known results in this direction. We use a new technique based on deep results of Talagrand from functional analysis.

Sampling discretization of the uniform norm and applications

arXiv (Cornell University), 2023

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.

The Marcinkiewicz-Type Discretization Theorems

Constructive Approximation

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.