{
  "id": "2105.07707",
  "version": "v2",
  "published": "2021-05-17T09:51:55.000Z",
  "updated": "2022-12-15T09:41:05.000Z",
  "title": "B-splines on the Heisenberg group",
  "authors": [
    "Santi R. Das",
    "Peter R. Massopust",
    "Radha Ramakrishnan"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "In this paper, we introduce a class of $B$-splines on the Heisenberg group $\\mathbb{H}$ and study their fundamental properties. Unlike the classical case, we prove that there does not exist any sequence $\\{\\alpha_n\\}_{n\\in\\mathbb{N}}$ such that $L_{(-n.-\\frac{n}{2},-\\alpha_n)}\\phi_n(x,y,t)=L_{(-n.-\\frac{n}{2},-\\alpha_n)}\\phi_n(-x,-y,-t)$, for $n\\geq 2$, where $L_{(x,y,t)}$ denotes the left translation on $\\mathbb{H}$. We further investigate the problem of finding an equivalent condition for the system of left translates to form a frame sequence or a Riesz sequence in terms of twisted translates. We also find a sufficient condition for obtaining an oblique dual of the system $\\{L_{(2k,l,m)}g:k,l,m\\in\\mathbb{Z}\\}$ for a certain class of functions $g\\in L^2(\\mathbb{H})$. These concepts are illustrated by some examples. Finally, we make some remarks about $B$-splines regarding these results.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-12-15T09:41:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C15",
      "41A15",
      "43A30"
    ],
    "keywords": [
      "heisenberg group",
      "sufficient condition",
      "riesz sequence",
      "fundamental properties",
      "frame sequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}