{
  "id": "2006.11581",
  "version": "v1",
  "published": "2020-06-20T14:09:10.000Z",
  "updated": "2020-06-20T14:09:10.000Z",
  "title": "Fourier transform on the Lobachevsky plane and operational calculus",
  "authors": [
    "Yury A. Neretin"
  ],
  "comment": "11p",
  "categories": [
    "math.RT",
    "math.CA",
    "math.FA"
  ],
  "abstract": "The classical Fourier transform on the line sends the operator of multiplication by $x$ to $i\\frac{d}{d\\xi}$ and the operator of differentiation $\\frac{d}{d x}$ to the multiplication by $-i\\xi$. For the Fourier transform on the Lobachevsky plane we establish a similar correspondence for a certain family of differential operators. It appears that differential operators on the Lobachevsky plane correspond to differential-difference operators in the Fourier-image, where shift operators act in the imaginary direction, i.e., a direction transversal to the integration contour in the Plancherel formula.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-20T14:09:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "43A85",
      "43A32",
      "44A99",
      "22E46"
    ],
    "keywords": [
      "operational calculus",
      "differential operators",
      "lobachevsky plane correspond",
      "shift operators act",
      "classical fourier transform"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}