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

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Approximation Theory, Tampa

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1287))

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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).

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Author information

Authors and Affiliations

  1. Department of Mathematics, Witwatersrand University, 1 Jan Smuts Avenue, 2001, Johannesburg, Republic of South Africa

    Arnold Knopfmacher

  2. National Research Institute for Mathematical Sciences, CSIR, P.O. Box 395, 0001, Pretoria, Republic of South Africa

    D. S. Lubinsky

Authors
  1. Arnold Knopfmacher
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  2. D. S. Lubinsky
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Edward B. Saff

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© 1987 Springer-Verlag

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Knopfmacher, A., Lubinsky, D.S. (1987). Analogues of Freud’s conjecture for Erdös type weights and related polynomial approximation problems. In: Saff, E.B. (eds) Approximation Theory, Tampa. Lecture Notes in Mathematics, vol 1287. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0078897

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  • DOI: https://doi.org/10.1007/BFb0078897

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  • Print ISBN: 978-3-540-18500-0

  • Online ISBN: 978-3-540-47991-8

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