{
  "id": "2010.05667",
  "version": "v1",
  "published": "2020-10-12T13:06:19.000Z",
  "updated": "2020-10-12T13:06:19.000Z",
  "title": "Frame spectral pairs and exponential bases",
  "authors": [
    "Christina Frederick",
    "Azita Mayeli"
  ],
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "Given a domain $\\Omega\\subset\\Bbb R^d$ with positive and finite Lebesgue measure and a discrete set $\\Lambda\\subset \\Bbb R^d$, we say that $(\\Omega, \\Lambda)$ is a {\\it frame spectral pair} if the set of exponential functions $\\mathcal E(\\Lambda):=\\{e^{2\\pi i \\lambda \\cdot x}: \\lambda\\in \\Lambda\\}$ is a frame for $L^2(\\Omega)$. Special cases of frames include Riesz bases and orthogonal bases.In the finite setting $\\Bbb Z_N^d$, $d, N\\geq 1$, a frame spectral pair can be defined in a similar way. %(Here, $\\Bbb Z_N$ is the cyclic abelian group of order.) We show how to construct and obtain a new frame spectral pair in $\\Bbb R^d$ by \"adding\" frame spectral pairs in $\\Bbb R^{d}$ and $\\Bbb Z_N^d$. Our construction unifies the well-known examples of exponential frames for the union of cubes with equal volumes. In this paper, we will also obtain a connection between frame spectral pairs and the Whittaker-Shannon interpolation formula when the frame is an orthogonal basis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-12T13:06:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "frame spectral pair",
      "exponential bases",
      "orthogonal basis",
      "finite lebesgue measure",
      "cyclic abelian group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}