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

Abstract

Let K⊂R be a regular compact set and let g(z)=gC¯∖K(z,∞) be the Green function for C¯∖K with pole at infinity. For δ>0, define G(δ)≔max{g(z):z∈C,dist(z,K)≤2δ}.Let {xn}n=0∞ be a Leja sequence of points of K. Then the uniform norm ‖Tn‖=Λn,n=1,2,… of the associated interpolation operator Tn, i.e., the nth Lebesgue constant, is bounded from above by minδ>02ndiam(K)δenG(δ)9/8.In particular, when K is a uniformly perfect subset of 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.