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

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 $S^q$, and investigate how well continuous $L_p$-norms of polynomials f of maximum degree n on the sphere $S^q$ can be discretized by positively weighted $L_p$-sum of finitely many samples, and discuss the distortion between the continuous and discrete quantities, the number and distribution of the (deterministic or randomly chosen) sample points ${\xi }_1,\dots ,{\xi }_N$ on $S^q$, the dimension q, and the degree n of the polynomials.