{
  "id": "2207.10706",
  "version": "v1",
  "published": "2022-07-21T18:40:50.000Z",
  "updated": "2022-07-21T18:40:50.000Z",
  "title": "From Schwartz space to Mellin transform",
  "authors": [
    "Mateusz Krukowski"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "The primary motivation behind this paper is an attempt to provide a thorough explanation of how the Mellin transform arises naturally in a process akin to the construction of the celebrated Gelfand transform. We commence with a study of a class of Schwartz functions $\\mathcal{S}(\\mathbb{R}_+),$ where $\\mathbb{R}_+$ is the set of all positive real numbers. Various properties of this Fr\\'echet space are established and what follows is an introduction of the Mellin convolution operator, which turns $\\mathcal{S}(\\mathbb{R}_+)$ into a commutative Fr\\'echet algebra. We provide a simple proof of Mellin-Young convolution inequality and go on to prove that the structure space $\\Delta(\\mathcal{S}(\\mathbb{R}_+),\\star)$ (the space of nonzero, linear, continuous and multiplicative functionals $m:\\mathcal{S}(\\mathbb{R}_+)\\longrightarrow \\mathbb{R}$) is homeomorphic to $\\mathbb{R}.$ Finally, we show that the Mellin transform arises in a process which bears a striking resemblance to the construction of the Gelfand transform.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-21T18:40:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "schwartz space",
      "mellin transform arises",
      "mellin convolution operator",
      "mellin-young convolution inequality",
      "simple proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}