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
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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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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DOI: https://doi.org/10.1007/BF02075447