{
  "id": "0910.4223",
  "version": "v1",
  "published": "2009-10-22T03:46:00.000Z",
  "updated": "2009-10-22T03:46:00.000Z",
  "title": "On the norms and roots of orthogonal polynomials in the plane and $L^p$-optimal polynomials with respect to varying weights",
  "authors": [
    "F. Balogh",
    "M. Bertola"
  ],
  "categories": [
    "math.CA",
    "math-ph",
    "math.MP"
  ],
  "abstract": "For a measure on a subset of the complex plane we consider $L^p$-optimal weighted polynomials, namely, monic polynomials of degree $n$ with a varying weight of the form $w^n = {\\rm e}^{-n V}$ which minimize the $L^p$-norms, $1 \\leq p \\leq \\infty$. It is shown that eventually all but a uniformly bounded number of the roots of the $L^p$-optimal polynomials lie within a small neighborhood of the support of a certain equilibrium measure; asymptotics for the $n$th roots of the $L^p$ norms are also provided. The case $p=\\infty$ is well known and corresponds to weighted Chebyshev polynomials; the case $p=2$ corresponding to orthogonal polynomials as well as any other $1\\leq p <\\infty$ is our contribution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-10-22T03:46:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "orthogonal polynomials",
      "varying weight",
      "optimal polynomials lie",
      "complex plane",
      "optimal weighted polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0910.4223B"
    }
  }
}