{
  "id": "2008.08079",
  "version": "v1",
  "published": "2020-08-18T17:59:25.000Z",
  "updated": "2020-08-18T17:59:25.000Z",
  "title": "Harmonic analysis of little $q$-Legendre polynomials",
  "authors": [
    "Stefan Kahler"
  ],
  "comment": "The paper is essentially also a part of the first version of arXiv:1806.00339 [math.FA]. It is now a separate paper because the associated symmetric Pollaczek part of arXiv:1806.00339 [math.FA] was extended. Compared to the (first version of) arXiv:1806.00339, we extended and added some results on little $q$-Legendre polynomials",
  "categories": [
    "math.FA"
  ],
  "abstract": "Many classes of orthogonal polynomials satisfy a specific linearization property giving rise to a polynomial hypergroup structure, which offers an elegant and fruitful link to harmonic and functional analysis. From the opposite point of view, this allows regarding certain Banach algebras as $L^1$-algebras, associated with underlying orthogonal polynomials or with the corresponding orthogonalization measures. The individual behavior strongly depends on these underlying polynomials. We study the little $q$-Legendre polynomials, which are orthogonal with respect to a discrete measure. Their $L^1$-algebras have been known to be not amenable but to satisfy some weaker properties like right character amenability. We will show that the $L^1$-algebras associated with the little $q$-Legendre polynomials share the property that every element can be approximated by linear combinations of idempotents. This particularly implies that these $L^1$-algebras are weakly amenable (i. e., every bounded derivation into the dual module is an inner derivation), which is known to be shared by any $L^1$-algebra of a locally compact group. As a crucial tool, we establish certain uniform boundedness properties of the characters. Our strategy relies on continued fractions, character estimations and asymptotic behavior.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-18T17:59:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33D45",
      "40A15",
      "43A20",
      "43A62"
    ],
    "keywords": [
      "harmonic analysis",
      "specific linearization property giving rise",
      "orthogonal polynomials satisfy",
      "right character amenability",
      "uniform boundedness properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}