{
  "id": "math/0407466",
  "version": "v1",
  "published": "2004-07-27T18:40:12.000Z",
  "updated": "2004-07-27T18:40:12.000Z",
  "title": "Fourier Coefficients of Beurling Functions and a Class of Mellin Transform Formally Determined by its Values on the Even Integers",
  "authors": [
    "F. Auil"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "It is a well-known fact that Riemann Hypothesis will follows if the function identically equal to -1 can be arbitrarily approximated in the norm $\\norma{.}$ of $L^{2}([0,1],dx)$ by functions of the form $f(x)=\\sum_{k=1}^{n}a_{k} \\rho(\\frac{\\theta_{k}}{x})$, where $\\rho(x)\\adef\\pfrac{x}$, and $a_{k}\\in\\cc$, $0<\\theta_{k}\\leqslant 1$ satisfies $\\sum_{k=1}^{n}a_{k} \\theta_{k}=0$. Parsevall Identity $\\norma{f(x)+1}^{2}=\\sum_{n\\in\\zz}\\modulo{c(n)}^{2}$ is a possible tool to compute or estimate this norm. In this note we give an expression for the Fourier coefficients $c(n)$ of $f+1$, when $f$ is a function defined as above. As an application, we derive an expression for $M_{f}(s)\\adef\\int_{0}^{1}(f(x)+1) x^{s-1} dx$ as a series that only depends on $M_{f}(2k)$, $k\\in\\nn$. We remark that the Fourier coefficients $c(n)$ depend on $M_{f}(2k)$ which, for a function $f$ defined as above, can be expressed also in terms of the $a_{k}$'s and $\\theta_{k}$'s. Therefore, a better control on these parameters will allow to estimate $M_{f}(2k)$ and therefore eventually to handle $\\norma{f+1}$ via our expression for the Fourier coefficients and Parsevall Identity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-07-27T18:40:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fourier coefficients",
      "mellin transform",
      "beurling functions",
      "parsevall identity",
      "expression"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......7466A"
    }
  }
}