On the computation of sets of points with low Lebesgue constant on the unit disk

Abstract

In this paper of numerical nature, we test the Lebesgue constant of several available point sets on the disk Ω and propose new ones that enjoy low Lebesgue constant. Furthermore we extend some results in Cuyt (2012), analyzing the case of Bos arrays whose radii are nonnegative Gauss–Gegenbauer–Lobatto nodes with exponent α , noticing that the optimal α still allows to achieve point sets on Ω with low Lebesgue constant Λ n for degrees n ≤ 30 . Next we introduce an algorithm that through optimization determines point sets with the best known Lebesgue constants for n ≤ 25 . Finally, we determine theoretically a point set with the best Lebesgue constant for the case n = 1 .