New bounds on the Lebesgue constants of Leja sequences on the unit disc and their projections R-Leja sequences

Motivated by the development of non-intrusive interpolation methods for parametric PDE’s in high dimension, we have introduced in [8] a sparse multi-variate polynomial interpolation procedure based on the Smolyak formula. The evaluation points lie in an infinite grid ⊗j=0Z where Z = (zj)j≥0 is any infinite sequence of mutually distinct points in some compact X in R or C. A key aspect of the interpolation procedure is its hierarchical structure: the sampling set is progressively enriched together with the polynomial space. The Lebesgue constant that quantifies the stability of the resulting interpolation operator depends on Z. We have shown in [6] that the interpolation operator has Lebesgue constant with quadratic and cubic growth in the number of points when Z is a Leja sequence on the unit disc initiated at the boundary and where Z is an R-Leja sequence on X = [−1, 1] respectively. This result followed from the linear and quadratic growth of the Lebesgue constant for univariate polynomial interpolation with the previous type of sequences, also established in [6]. In this paper, we derive new properties of these sequences and new bounds on the growth of the Lebesgue constants which improve those obtained in [6].