{
  "id": "1211.2633",
  "version": "v1",
  "published": "2012-11-12T14:29:47.000Z",
  "updated": "2012-11-12T14:29:47.000Z",
  "title": "Step refinable functions and orthogonal MRA on $p$-adic Vilenkin groups",
  "authors": [
    "S. F. Lukomskii"
  ],
  "comment": "24 page",
  "categories": [
    "math.FA"
  ],
  "abstract": "We find the necessary and sufficient conditions for refinable step function under which this function generates an orthogonal MRA in the $L_2(\\mathfrak G)$ -spaces on Vilenkin groups $\\mathfrak G$. We consider a class of refinable step functions for which the mask $m_0(\\chi)$ is constant on cosets $\\mathfrak G_{-1}^\\bot$ and its modulus $|m_0(\\chi)|$ takes two values only: 0 and 1. We will prove that any refinable step function $\\varphi$ from this class that generates an orthogonal MRA on $p$-adic Vilenkin group $\\mathfrak G$ has Fourier transform with condition ${\\rm supp} \\hat \\varphi(\\chi)\\subset \\mathfrak G_{p-2}^\\bot$. We show the sharpness of this result too.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-11-12T14:29:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "adic vilenkin group",
      "orthogonal mra",
      "step refinable functions",
      "refinable step function",
      "function generates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.2633L"
    }
  }
}