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

2023/02/28 by Frank Filbir, Filbir, Frank, Ralf Hielscher +5 · 3 citations
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical functions and polynomials #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2303.00045

openalex publication_date 2023/02/28 · openalex created_date 2023/03/03 · openalex updated_date 2026/07/28

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 \mathbbS^q, and investigate how well continuous L_p-norms of polynomials f of maximum degree n on the sphere \mathbbS^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 ξ₁,…,ξ_N on \mathbbS^q, the dimension q, and the polynomial degree n.

Cited by

• Beyond Tchakaloff Quadrature: Positive Functionals, Frames and Widths
• The path of hyperinterpolation: A survey
• Semi-discrete moduli of smoothness and their applications in one- and two- sided error estimates

Related

• Approximation and quadrature by weighted least squares polynomials on the sphere
• Marcinkiewicz--Zygmund inequalities for scattered data on polygons
• Weighted ℓ_q approximation problems on the ball and on the sphere
• Weighted L^p bounds for the Marcinkiewicz integral
• Compressive sensing and truncated moment problems on spheres