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Marcinkiewicz--Zygmund inequalities for scattered data on polygons

Hao-Ning Wu

TL;DR

This work addresses constructing positive-weight quadrature rules for polygonal domains using scattered data to establish Marcinkiewicz--Zygmund inequalities for $1\le p\le \infty$. The authors triangulate the polygon and build a degree-$d$ exact quadrature on triangles via Bernstein--Bézier polynomials, obtaining triangle weights $w_{ijk}$ and aggregating to polygon weights $w_j$ to achieve MZ-type control on polygons. They provide exactness for polynomials up to degree $d$ on triangles, derive MZ inequalities for $N\le d$ (and $p=\infty$ via analogous bounds), and extend the framework to polygons with mesh-density dependent conditions and constants tied to the boundary geometry, supported by error analyses and numerical tests. The results yield a practical, geometry-aware approach that relaxes strict quadrature exactness while preserving stable $L^p$ control for polygonal domains with scattered data.

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

Marcinkiewicz--Zygmund inequalities for scattered data on polygons

TL;DR

This work addresses constructing positive-weight quadrature rules for polygonal domains using scattered data to establish Marcinkiewicz--Zygmund inequalities for . The authors triangulate the polygon and build a degree- exact quadrature on triangles via Bernstein--Bézier polynomials, obtaining triangle weights and aggregating to polygon weights to achieve MZ-type control on polygons. They provide exactness for polynomials up to degree on triangles, derive MZ inequalities for (and via analogous bounds), and extend the framework to polygons with mesh-density dependent conditions and constants tied to the boundary geometry, supported by error analyses and numerical tests. The results yield a practical, geometry-aware approach that relaxes strict quadrature exactness while preserving stable control for polygonal domains with scattered data.

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 . As a byproduct, we provide error analysis for both quadrature rules on triangles and polygons. Numerical results further validate our construction.

Paper Structure

This paper contains 7 sections, 12 theorems, 92 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. On triangles
  5. On polygons
  6. Numerical results
  7. Final remark

Key Result

Lemma 2.1

Let $P$ be a polynomial in the B-form equ:Bform with coefficient vector $\bm{c}$. Then and for $1\leq p<\infty$, where $K_0>0$ is a constant depending only on $d$, and $K_0 = 1$ when $p=\infty$ and $d=1$.

Figures (2)

  • Figure 1: Locations of domain points and corresponding weights for various $d$.
  • Figure 2: Locations of quadrature points, corresponding quadrature weights, and triangulation.

Theorems & Definitions (13)

  • Lemma 2.1: Norm equivalence laischumakr
  • Lemma 2.2: Interpolation error zbMATH01234311
  • Lemma 2.3: Markov laischumakr
  • theorem 1: Quadrature on triangles: exactness
  • theorem 2: Quadrature on triangles: error analysis
  • theorem 3: MZ on triangles, I
  • theorem 4: MZ-like on triangles, II
  • Remark 3.1
  • Lemma 3.1: Markov on polygons
  • Lemma 3.2
  • ...and 3 more