Marcinkiewicz--Zygmund inequalities for scattered data on polygons

2024/11/25 by Hao-Ning Wu, Wu, Hao-Ning · 1 citation
Mathematics · #Point processes and geometric inequalities #Advanced Harmonic Analysis Research #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2411.16584

Abstract

Given a set of scattered points on a regular or irregular 2D polygon, we aim to employ them as quadrature points to construct a quadrature rule that establishes Marcinkiewicz--Zygmund inequalities on this polygon. The quadrature construction is aided by Bernstein--Bézier polynomials. For this purpose, we first propose a quadrature rule on triangles with an arbitrary degree of exactness and establish Marcinkiewicz--Zygmund estimates for 3-, 10-, and 21-point quadrature rules on triangles. Based on the 3-point quadrature rule on triangles, we then propose the desired quadrature rule on the polygon that satisfies Marcinkiewicz--Zygmund inequalities for 1≤ p ≤ ∞. As a byproduct, we provide error analysis for both quadrature rules on triangles and polygons. Numerical results further validate our construction.

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