{
  "id": "2408.09099",
  "version": "v1",
  "published": "2024-08-17T05:00:43.000Z",
  "updated": "2024-08-17T05:00:43.000Z",
  "title": "Construction of irregular complete interpolation sets for shift-invariant spaces",
  "authors": [
    "Kumari Priyanka",
    "A. Antony Selvan"
  ],
  "comment": "30 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "For several shift-invariant spaces, there exists a real number $a\\in\\mathbb{R}$ such that the set $a+\\mathbb{Z}$ is a complete interpolation set. In this paper, we characterize the complete interpolation property of the set $(a+\\mathbb{N}_0)\\cup(\\alpha+a+\\mathbb{N}^{-})$ for shift-invariant spaces using Toeplitz operators. Using this characterization, we determine all $\\alpha$ for which the sample set $\\mathbb{N}_0\\cup\\alpha+\\mathbb{N}^{-}$ forms a complete interpolation set for transversal-invariant spaces. We introduce a new recurrence relation for exponential splines, examines the zeros of these splines, and explores the zero-free region of the doubly infinite Lerch zeta function. Consequently, we demonstrate that $\\left\\langle\\frac{m}{2}\\right\\rangle+\\mathbb{N}_0\\cup\\alpha+\\left\\langle\\frac{m}{2}\\right\\rangle+\\mathbb{N}^{-}$ is a complete interpolation set for a shift-invariant spline space of order $m\\geq 2$ if and only if $|\\alpha|<1/2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-17T05:00:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "irregular complete interpolation sets",
      "shift-invariant spaces",
      "construction",
      "doubly infinite lerch zeta function",
      "complete interpolation property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}