Weak limits of zeros of orthogonal polynomials
Let \(P_ n(x,\mu)\) be the monic orthogonal polynomial associated with the positive unit Borel measure \(\mu\) with infinite support on \(I=[-1,1]\) and let \(\nu_ n=\nu_ n(\mu)\) denote the zero measure of \(P_ n(x,\mu)\), i.e., the unit measure with mass 1/n at each zero of \(P_ n(x,\mu)\). The sequence \(\{\nu_ n\}^{\infty}_{n=1}\) converges weakly to the measure \(\nu\) if \(\lim_{n\to \infty}\int fd\nu_ n=\int fd\nu\) for all functions f(x) continuous on I. A carrier is defined as a Borel subset of the support having unit measure and two measures are called carrier related if they share the same set of carriers. The authors demonstrate that the equilibrium measures are weak limits of the zero measures of the orthogonal polynomials associated with the carrier related measures.
scientific article; zbMATH DE number 3979731
scientific article