{
  "id": "2402.05230",
  "version": "v1",
  "published": "2024-02-07T20:09:59.000Z",
  "updated": "2024-02-07T20:09:59.000Z",
  "title": "Mittag-Leffler functions in the Fourier space",
  "authors": [
    "Ahmed A. Abdelhakim"
  ],
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "Let $\\alpha \\in (0,2)$ and let $\\beta>0$. Fix $-\\pi<\\varphi\\leq \\pi$ such that $|\\varphi|>\\alpha \\pi/2$. We determine the precise asymptotic behaviour of the Fourier transform of $E_{\\alpha,\\beta}(e^{\\dot{\\imath} \\varphi} |\\cdot|^{\\sigma})$ whenever $\\sigma>(n-1)/2$. Remarkably, this asymptotic behaviour turns out to be independent of $\\alpha$ and $\\beta$. This helps us determine the values of the Lebesgue exponent $p=p(\\sigma)$, $\\sigma>(n-1)/2$, for which $\\mathcal{F} \\left(E_{\\alpha,\\beta}(e^{\\dot{\\imath} \\varphi} |\\cdot|^{\\sigma})\\right)$ is in $L^{p}(\\mathbb{R}^{n})$. These values cannot be obtained via the Hausdorff-Young inequality. This problem arises in the study of space-time fractional equations. Our approach provides an effective alternative to the asymptotic analysis of the Fox $H-$ functions recently applied to the case $\\alpha \\in (0,1)$, $\\beta=\\alpha, 1$, $\\varphi=-\\pi/2,\\pi$. We rather rely on an appropriate integral representative that represents $E_{\\alpha,\\beta}$ continuously up to the origin, and we develop an extended asymptotic expansion for the Bessel function whose coefficients and remainder term are obtained explicitly.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-02-07T20:09:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30E15",
      "33E12",
      "42B10",
      "30E20",
      "34E05"
    ],
    "keywords": [
      "mittag-leffler functions",
      "fourier space",
      "asymptotic behaviour turns",
      "precise asymptotic behaviour",
      "space-time fractional equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}