{
  "id": "1211.1187",
  "version": "v1",
  "published": "2012-11-06T11:31:43.000Z",
  "updated": "2012-11-06T11:31:43.000Z",
  "title": "Interpolation, box splines, and lattice points in zonotopes",
  "authors": [
    "Matthias Lenz"
  ],
  "comment": "10 pages, 3 figures",
  "doi": "10.1093/imrn/rnt142",
  "categories": [
    "math.CO",
    "math.AC",
    "math.NA"
  ],
  "abstract": "Let $X$ be a totally unimodular list of vectors in some lattice. Let $B_X$ be the box spline defined by $X$. Its support is the zonotope $Z(X)$. We show that any real-valued function defined on the set of lattice points in the interior of $Z(X)$ can be extended to a function on $Z(X)$ of the form $p(D)B_X$ in a unique way, where $p(D)$ is a differential operator that is contained in the so-called internal $\\Pcal$-space. This was conjectured by Olga Holtz and Amos Ron. We also point out connections between this interpolation problem and matroid theory, including a deletion-contraction decomposition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-11-06T11:31:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B35",
      "41A05",
      "41A15",
      "52B20",
      "13F20",
      "41A63",
      "47F05",
      "52B40",
      "52C07"
    ],
    "keywords": [
      "lattice points",
      "deletion-contraction decomposition",
      "totally unimodular list",
      "differential operator",
      "olga holtz"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.1187L"
    }
  }
}