On the Lebesgue constant of Leja sequences for the complex unit disk and of their real projection

We consider Leja sequences of points for polynomial interpolation on the complex unit disk UU and the corresponding sequences for polynomial interpolation on the real interval [−1,1][−1,1] obtained by projection. It was proved by Calvi and Phung in Calvi and Phung (2011, 2012) [3] and [4] that the Lebesgue constants for such sequences are asymptotically bounded in O(klogk)O(klogk) and O(k3logk)O(k3logk) respectively, where kk is the number of points. In this paper, we establish the sharper bound 5k2logk5k2logk in the real interval case. We also give sharper bounds in the complex unit disk case, in particular 2k2k. Our motivation for producing such sharper bounds is the use of these sequences in the framework of adaptive sparse polynomial interpolation in high dimension.