Applications of universality limits to zeros and reproducing kernels of orthogonal polynomials

  • Authors:

  • Eli Levin;Doron S. Lubinsky

  • Affiliations:

  • Mathematics Department, The Open University of Israel, P.O. Box 808, Raanana 43107, Israel;School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2008

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Abstract

We apply universality limits to asymptotics of spacing of zeros x"k"n of orthogonal polynomials, for weights with compact support and for exponential weights. A typical result islimn-~x"k"n-x"k"+"1","nK@?"nx"k"n,x"k"n=1under minimal hypotheses on the weight, with K@?"n denoting a normalized reproducing kernel. Moreover, for exponential weights, we derive asymptotics for the differentiated kernels:K"n^r^,^sx,x=@?k=0n-1p"k^rxp"k^sx.