Marcinkiewicz--Zygmund inequalities for scattered and random data on the $q$-sphere

TL;DR

This paper studies how well continuous norms of polynomials on the $q$-sphere can be approximated by discrete sums over scattered or random points, with implications for function recovery and integration.

Contribution

It extends Marcinkiewicz--Zygmund inequalities to the $q$-sphere for scattered and random data, analyzing discretization accuracy in relation to sampling strategies.

Findings

Discretization bounds depend on sample distribution and number.

Results apply to both deterministic and random sampling.

Provides theoretical foundations for polynomial approximation on spheres.

Abstract

The recovery of multivariate functions and estimating their integrals from finitely many samples is one of the central tasks in modern approximation theory. Marcinkiewicz--Zygmund inequalities provide answers to both the recovery and the quadrature aspect. In this paper, we put ourselves on the $q$-dimensional sphere $\mathbb{S}^q$, and investigate how well continuous $L_p$-norms of polynomials $f$ of maximum degree $n$ on the sphere $\mathbb{S}^q$ can be discretized by positively weighted $L_p$-sum of finitely many samples, and discuss the relationship between the offset between the continuous and discrete quantities, the number and distribution of the (deterministic or randomly chosen) sample points $\xi_1,\ldots,\xi_N$ on $\mathbb{S}^q$, the dimension $q$, and the polynomial degree $n$.