Marcinkiewicz--Zygmund inequalities for scattered data on polygons

TL;DR

This paper develops quadrature rules for scattered data on polygons using Bernstein--Bézier polynomials, establishing Marcinkiewicz--Zygmund inequalities and providing error analysis with numerical validation.

Contribution

It introduces a novel quadrature rule on polygons based on triangle rules and Bernstein--Bézier polynomials, establishing Marcinkiewicz--Zygmund inequalities for scattered data.

Findings

Established Marcinkiewicz--Zygmund inequalities for 3-, 10-, and 21-point triangle quadrature rules.

Constructed a quadrature rule on polygons satisfying Marcinkiewicz--Zygmund inequalities for 1 ≤ p ≤ ∞.

Validated the quadrature construction with numerical experiments.

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--B\'{e}zier 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 $1\leq p \leq \infty$. As a byproduct, we provide error analysis for both quadrature rules on triangles and polygons. Numerical results further validate our construction.