{
  "id": "2210.06076",
  "version": "v1",
  "published": "2022-10-12T10:41:59.000Z",
  "updated": "2022-10-12T10:41:59.000Z",
  "title": "Discrete Analogues in Harmonic Analysis: A Theorem of Stein-Wainger",
  "authors": [
    "Ben Krause"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "For $d \\geq 2, \\ D \\geq 1$, let $\\mathscr{P}_{d,D}$ denote the set of all degree $d$ polynomials in $D$ dimensions with real coefficients without linear terms. We prove that for any Calder\\'{o}n-Zygmund kernel, $K$, the maximally modulated and maximally truncated discrete singular integral operator, \\begin{align*} \\sup_{P \\in \\mathscr{P}_{d,D}, \\ N} \\Big| \\sum_{0 < |m| \\leq N} f(x-m) K(m) e^{2\\pi i P(m)} \\Big|, \\end{align*} is bounded on $\\ell^p(\\mathbb{Z}^D)$, for each $1 < p < \\infty$. Our proof introduces a stopping time based off of equidistribution theory of polynomial orbits to relate the analysis to its continuous analogue, introduced and studied by Stein-Wainger: \\begin{align*} \\sup_{P \\in \\mathscr{P}_{d,D}} \\Big| \\int_{\\mathbb{R}^D} f(x-t) K(t) e^{2\\pi i P(t)} \\ dt \\Big|. \\end{align*}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-12T10:41:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "harmonic analysis",
      "discrete analogues",
      "stein-wainger",
      "truncated discrete singular integral operator",
      "maximally truncated discrete singular integral"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}