{
  "id": "1505.01801",
  "version": "v1",
  "published": "2015-05-07T18:12:42.000Z",
  "updated": "2015-05-07T18:12:42.000Z",
  "title": "Simplex Spline Bases on the Powell-Sabin 12-Split: Part II",
  "authors": [
    "Tom Lyche",
    "Georg Muntingh"
  ],
  "comment": "Oberwolfach report for the conference Multivariate Splines and Algebraic Geometry",
  "categories": [
    "math.NA"
  ],
  "abstract": "For the space $\\mathcal{S}$ of $C^3$ quintics on the Powell-Sabin 12-split of a triangle, we determine the simplex splines in $\\mathcal{S}$ and the six symmetric simplex spline bases that reduce to a B-spline basis on each edge, have a positive partition of unity, a (barycentric) Marsden identity, and domain points with an intuitive control net. We provide a quasi-interpolant with approximation order 6 and a Lagrange interpolant at the domain points. The latter can be used to show that each basis is stable in the $L_\\infty$ norm, which yields an $h^2$ bound for the distance between the B\\'ezier ordinates and the values of the spline at the corresponding domain points. Finally, for one of these bases we provide $C^0$, $C^1$, and $C^2$ conditions on the control points of two splines on adjacent macrotriangles, and a conversion to the Hermite nodal basis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-07T18:12:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A15",
      "65D07",
      "65D17"
    ],
    "keywords": [
      "powell-sabin",
      "symmetric simplex spline bases",
      "hermite nodal basis",
      "bezier ordinates",
      "lagrange interpolant"
    ],
    "tags": [
      "conference paper"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150501801L"
    }
  }
}