A simple upper bound for Lebesgue constants associated with Leja points on the real line

2021/11/08 by Andrievskii, Vladimir, Nazarov, Fedor
#41A05 #41A10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2111.04607

Abstract

Let K⊂ \mathbb R be a regular compact set and let g(z)=g_\mathbb C∖ K(z,∞) be the Green function for \mathbb C∖ K with pole at infinity. For δ>0, define G(δ):=max\ g(z): z∈ \mathbb C, dist(z,K)≤ 2δ\. Let \ x_n\_n=0^∞ be a Leja sequence of points of K. Then the uniform norm ‖T_n‖=Λ_n, n=1,2,… of the associated interpolation operator T_n, i.e., the n-th Lebesgue constant, is bounded from above by min_δgt;02n[\fracdiam( K)δe^nG(δ)]^9/8. In particular, when K is a uniformly perfect subset of \mathbb R, the Lebesgue constants grow at most polynomially in n. To the best of our knowledge, the result is new even when K is a finite union of intervals.

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