The Marcinkiewicz-type discretization theorems for the hyperbolic cross polynomials

TL;DR

This paper investigates the discretization of hyperbolic cross trigonometric polynomials, addressing a complex problem using probabilistic and number theoretical methods to advance understanding in this area.

Contribution

It introduces two novel techniques—probabilistic and number theoretical—for studying discretization theorems for hyperbolic cross polynomials.

Findings

Development of probabilistic and number theoretical approaches

Initial results on discretization theorems for hyperbolic cross polynomials

Foundation for further systematic study of the problem

Abstract

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.