Related papers: The Marcinkiewicz-type discretization theorems
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…
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…
The paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. We use recent general results on sampling discretization to derive a new Marcinkiewicz type discretization theorem for the…
This paper surveys recent developments in the sampling discretization of integral and uniform norms for functions in general finite-dimensional spaces. These results generalize the classical Marcinkiewicz-Zygmund inequalities for…
The paper studies the sampling discretization problem for integral norms on subspaces of $L^p(\mu)$. Several close to optimal results are obtained on subspaces for which certain Nikolskii-type inequality is valid. The problem of norms…
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 --…
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 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…
This survey addresses sampling discretization and its connections with other areas of mathematics. The survey concentrates on sampling discretization of norms of elements of finite-dimensional subspaces. We present here known results on…
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…
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…
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,…
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…
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…
As an alternative to the well-known methods of "chaining" and "bracketing" that have been developed in the study of random fields, a new method, which is based on a {\em stochastic maximal inequality} derived by using the formula for…
We prove a sampling discretization theorem for the square norm of functions from a finite dimensional subspace satisfying Nikol'skii's inequality with an upper bound on the number of sampling points of the order of the dimension of the…
$D$-optimal designs originate in statistics literature as an approach for optimal experimental designs. In numerical analysis points and weights resulting from maximal determinants turned out to be useful for quadrature and interpolation.…
We obtain new sampling discretization results in Orlicz norms on finite dimensional spaces. As applications, we study sampling recovery problems, where the error of the recovery process is calculated with respect to different Orlicz norms.…
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…
This chapter describes techniques for the numerical resolution of optimal transport problems. We will consider several discretizations of these problems, and we will put a strong focus on the mathematical analysis of the algorithms to solve…