Menke points on the real line and their connection to classical orthogonal polynomials

摘要

We investigate the properties of extremal point systems on the real line consisting of two interlaced sets of points solving a modified minimum energy problem. We show that these extremal points for the intervals [−1,1], [0,∞) and (−∞,∞), which are analogues of Menke points for a closed curve, are related to the zeros and extrema of classical orthogonal polynomials. Use of external fields in the form of suitable weight functions instead of constraints motivates the study of “weighted Menke points” on [0,∞) and (−∞,∞). We also discuss the asymptotic behavior of the Lebesgue constant for the Menke points on [−1,1].