Orthogonal polynomials of equilibrium measures supported on Cantor sets

Abstract

The equilibrium measure of a compact set is a fundamental object in logarithmic potential theory. We compute numerically this measure and its orthogonal polynomials, when the compact set is a Cantor set, defined by an Iterated Function System.We first construct sequences of discrete measures, via the solution of large systems of non-linear equations, that converge weakly to the equilibrium measure. Successively, we compute their Jacobi matrices via standard procedures, enhanced for the scope. Numerical estimates of the convergence rate to the limit Jacobi matrix are provided, which show stability and efficiency of the whole procedure. As a companion result, we also compute Jacobi matrices in two other cases: equilibrium measures on finite sets of intervals, and balanced measures of Iterated Function Systems.These algorithms can reach large polynomial orders: therefore, we study the asymptotic behavior of the orthogonal polynomials and, by a natural extension of the concept of regular root asymptotics, we derive an efficient scheme for the computation of complex Green’s functions and of related conformal mappings.