{
  "id": "2411.16584",
  "version": "v2",
  "published": "2024-11-25T17:14:20.000Z",
  "updated": "2024-11-28T15:59:24.000Z",
  "title": "Marcinkiewicz--Zygmund inequalities for scattered data on polygons",
  "authors": [
    "Hao-Ning Wu"
  ],
  "comment": "v2: 16 pages; corrected a critical typo in Theorem 3.6 and added a new Corollary 3.1",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-11-28T15:59:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A17",
      "41A05",
      "65D32",
      "42C15"
    ],
    "keywords": [
      "scattered data",
      "satisfies marcinkiewicz-zygmund inequalities",
      "irregular 2d polygon",
      "establishes marcinkiewicz-zygmund inequalities",
      "establish marcinkiewicz-zygmund estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}