{
  "id": "2102.05857",
  "version": "v1",
  "published": "2021-02-11T06:15:34.000Z",
  "updated": "2021-02-11T06:15:34.000Z",
  "title": "Nearly outer functions as extreme points in punctured Hardy spaces",
  "authors": [
    "Konstantin M. Dyakonov"
  ],
  "comment": "18 pages",
  "categories": [
    "math.FA",
    "math.CA",
    "math.CV"
  ],
  "abstract": "The Hardy space $H^1$ consists of the integrable functions $f$ on the unit circle whose Fourier coefficients $\\widehat f(k)$ vanish for $k<0$. We are concerned with $H^1$ functions that have some additional (finitely many) holes in the spectrum, so we fix a finite set $\\mathcal K$ of positive integers and consider the \"punctured\" Hardy space $$H^1_{\\mathcal K}:=\\{f\\in H^1:\\,\\widehat f(k)=0\\,\\,\\,\\text{for all }\\, k\\in\\mathcal K\\}.$$ We then investigate the geometry of the unit ball in $H^1_{\\mathcal K}$. In particular, the extreme points of the ball are identified as those unit-norm functions in $H^1_{\\mathcal K}$ which are not too far from being outer (in the appropriate sense). This extends a theorem of de Leeuw and Rudin that deals with the classical $H^1$ and characterizes its extreme points as outer functions. We also discuss exposed points of the unit ball in $H^1_{\\mathcal K}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-11T06:15:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30H10",
      "30J10",
      "42A32",
      "46A55"
    ],
    "keywords": [
      "extreme points",
      "punctured hardy spaces",
      "outer functions",
      "unit ball",
      "unit circle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}