Some improved bounds in sampling discretization of integral norms

TL;DR

This paper improves bounds on the number of sample points needed for discretizing integral norms of functions in finite-dimensional subspaces, using advanced functional analysis techniques.

Contribution

It introduces new bounds for sampling discretization of $L_p$ norms, leveraging Talagrand's deep results and under standard Nikol'skii-type assumptions.

Findings

Improved upper bounds on sample points for $L_p$ norm discretization.

Applicable to finite-dimensional subspaces of continuous functions.

Utilizes novel techniques based on Talagrand's functional analysis results.

Abstract

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, $1\le p<2$, 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.