{
  "id": "2009.12772",
  "version": "v1",
  "published": "2020-09-27T07:29:45.000Z",
  "updated": "2020-09-27T07:29:45.000Z",
  "title": "Pointwise Convergence of Fourier Series on the Ring of Integers of Local Fields with an Application to Gabor Systems",
  "authors": [
    "Md Nurul Molla",
    "Biswaranjan Behera"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We construct a simple example of an integrable function on the ring of integers of the $p$-adic field $\\Q_p$ having an almost everywhere divergent Fourier series. On the other hand, we prove the pointwise convergence of the Fourier series of functions in $L^p(\\D,w)$, $1<p<\\infty$, where $\\D$ is the ring of integers of a local field $K$ and $w$ is a weight in the Muckenhoupt $A_p$ class. This result includes, as special cases, when $\\D$ is the ring of integers of $\\Q_p$ or the field $\\mathbb{F}_q((X))$ of formal Laurent series over a finite field $\\mathbb{F}_q$, and in particular, when $\\D$ is the Walsh-Paley or dyadic group $2^\\omega$. To achieve this, we establish a weighted estimate for the maximal operator corresponding to the Fourier partial sum operators for functions in $L^p(\\D,w)$. As an application, we characterize the Schauder basis property of the Gabor systems in a local field $K$ of positive characteristic in terms of the $A_2$ weights on $\\D\\times\\D$ and the Zak transform $Zg$ of the window function $g$ that generates the Gabor system. Some examples are given to illustrate this result. In particular, we construct an example of a Gabor system which is complete and minimal, but fails to be a Schauder basis for $L^2(K)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-27T07:29:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "43A70",
      "43A50",
      "43A25"
    ],
    "keywords": [
      "gabor system",
      "local field",
      "pointwise convergence",
      "application",
      "fourier partial sum operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}