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Uniform and mean approximation by certain weighted polynomials, with applications

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Abstract

LetW(x):= exp(-{tiQ(x})), where, for example, Q(x) is even and convex onR, and Q(x)/logx → ∞ asx → ∞. A result of Mhaskar and Saff asserts that ifa n =a n (W) is the positive root of the equation

$$n = ({2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi })\int_0^1 {{{a_n xQ'(a_n x)} \mathord{\left/ {\vphantom {{a_n xQ'(a_n x)} {\sqrt {1 - x^2 } }}} \right. \kern-\nulldelimiterspace} {\sqrt {1 - x^2 } }}dx,}$$

then, given any polynomialP n(x) of degree at mostn, the sup norm ofP n(x)W(a n x) overR is attained on [-1, 1]. In addition, any sequence of weighted polynomials {p n (x)W(a n x)} ∞1 that is uniformly bounded onR will converge to 0, for ¦x¦>1.

In this paper we show that under certain conditions onW, a function g(x) continuous inR can be approximated in the uniform norm by such a sequence {p n (x)W(a n x)} ∞1 if and only if g(x)=0 for ¦x¦⩾ 1. We also prove anL p analogue for 0<p<∞. Our results confirm a conjecture of Saff forW(x)=exp(−|x|α), when α >1. Further applications of our results are upper bounds for Christoffel functions, and asymptotic behavior of the largest zeros of orthogonal polynomials. A final application is an approximation theorem that will be used in a forthcoming proof of Freud's conjecture for |x|p exp(−|x|α),α > 0,p > −1.

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Authors and Affiliations

  1. Numerical and Applied Mathematics Division, National Research Institute for Mathematical Sciences CSIR, P.O. Box 395, 0001, Pretoria, Republic of South Africa

    D. S. Lubinsky

  2. Institute for Constructive Mathematics Department of Mathematics, University of South Florida, 33620, Tampa, Florida, USA

    E. B. Saff

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  1. D. S. Lubinsky
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Communicated by Paul Nevai.

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Lubinsky, D.S., Saff, E.B. Uniform and mean approximation by certain weighted polynomials, with applications. Constr. Approx 4, 21–64 (1988). https://doi.org/10.1007/BF02075447

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