{
  "id": "1901.08713",
  "version": "v1",
  "published": "2019-01-25T02:25:09.000Z",
  "updated": "2019-01-25T02:25:09.000Z",
  "title": "Polynomials on the Sierpinski Gasket with Respect to Different Laplacians which are Symmetric and Self-Similar",
  "authors": [
    "Christian Loring",
    "W. Jacob Ogden",
    "Ely Sandine",
    "Robert S. Strichartz"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "We study the analogue of polynomials (solutions to $\\Delta u^{n+1} =0$ for some $n$) on the Sierpinski gasket ($SG$) with respect to a family of symmetric, self-similar Laplacians constructed by Fang, King, Lee, and Strichartz, extending the work of Needleman, Strichartz, Teplyaev, and Yung on the polynomials with respect to the standard Kigami Laplacian. We define a basis for the space of polynomials, the monomials, characterized by the property that a certain \"derivative\" is 1 at one of the boundary points, while all other \"derivatives\" vanish, and we compute the values of the monomials at the boundary points of $SG$. We then present some data which suggest surprising relationships between the values of the monomials at the boundary and certain Neumann eigenvalues of the family of symmetric self-similar Laplacians. Surprisingly, the results for the general case are quite different from the results for the Kigami Laplacian.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-25T02:25:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A80"
    ],
    "keywords": [
      "sierpinski gasket",
      "polynomials",
      "boundary points",
      "standard kigami laplacian",
      "symmetric self-similar laplacians"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}