Two examples based on the properties of discrete measures

TL;DR

The paper presents two examples illustrating properties of discrete measures, including a construction of measures with specific orthogonal polynomial and convergence properties related to logarithmic potentials.

Contribution

It introduces a method to construct discrete measures with prescribed support and convergence characteristics, based on weighted Leja points and properties of equilibrium measures.

Findings

Existence of a discrete measure with polynomial zero counting measures converging to a given measure.

An example of measures supported on a compact set that converge weak-* but not in potential to the equilibrium measure.

Demonstration of complex convergence behavior of logarithmic potentials for discrete measures.

Abstract

In the paper we represent two examples which are based on the properties of discrete measures. In the first part of the paper we prove that for each probability measure $\mu$, $\operatorname{supp}{\mu}=[-1,1]$, which logarithmic potential is a continuous function on $[-1,1]$ there exists a (discrete) measure $\sigma=\sigma(\mu)$, $\operatorname{supp}{\sigma}=[-1,1]$, with the following property. Let $\{P_n(x;\sigma)\}$ be the sequence of polynomials orthogonal with respect to $\sigma$. Then $\dfrac1n\chi(P_n(\cdot;\sigma))\overset{*}\to\mu$, $n\to\infty$, where $\chi(\cdot)$ is zero counting measure for the corresponding polynomial. The construction of the measure $\sigma$ is based of the properties of weighted Leja points. In the second part we give an example of a compact set and a sequence of discrete measures supported on that compact set with the following property. The sequence of measures converges in weak-$*$ topology to the equilibrium measure for the compact set but the corresponding sequence of the logarithmic potentials does not converge to the equilibrium potential in any neighbourhood of the compact set.