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Analogues of Freud’s conjecture for Erdös type weights and related polynomial approximation problems

  • Arnold Knopfmacher
  • D. S. Lubinsky
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1287)

Abstract

Let W(x):=e−Q(x), where Q(x)→∞ as |x|→∞ faster than any polynomial. Erdös [3] investigated orthogonal polynomials for weights of this type. Here we obtain asymptotics for the associated recurrence relation coefficients, analogous to those obtained recently for weights such as exp(−|x|α), α>0. Our results apply to weights such as W(x) ≔exp(−exp(|x|α)) or W(x) ≔exp(−exp(exp(|x|α))), α>0 arbitrary.

As a preliminary step, we investigate the possibility of approximation on the real line by weighted polynomials of the form Pn(x)W(anx), where Pn(x) is of degree at most n, and {an} 1 is a certain increasing sequence of positive numbers. Further, we investigate the asymptotic behaviour of entire functions that have nonnegative Maclaurin series coefficients, and that are associated with W2(x).

AMS (MOS) Classification

Primary 42C05 Secondary 41A10 

Keywords

Orthogonal Polynomials Asymptotics Recurrence Relation Coefficients Freud’s Conjecture Erdös Weights 

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Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Arnold Knopfmacher
    • 1
  • D. S. Lubinsky
    • 2
  1. 1.Department of MathematicsWitwatersrand UniversityJohannesburgRepublic of South Africa
  2. 2.National Research Institute for Mathematical Sciences, CSIRPretoriaRepublic of South Africa

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