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

02/28/2023

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by   Frank Filbir, et al.

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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 relationship between the offset between the continuous and discrete quantities, the number and distribution of the (deterministic or randomly chosen) sample points ξ_1,…,ξ_N on 𝕊^q, the dimension q, and the polynomial degree n.