Zero distributions via orthogonality. (Q2566833)

Let \(\mu\) be a finite positive Borel measure with infinite compact support \(S\subset\mathbb R\) and consider the monic orthogonal polynomials \(q_n(x)=x^n+\cdots\) satisfying \[ \int q_n(t)t^k d\mu(t)=0,\quad\quad k=0,1,\ldots,n-1. \] A known result states that if \(S\) is regular with respect to the Dirichlet problem in \(\overline{\mathbb{C}}\setminus S\) and if \(\mu\) is ``sufficiently thick,'' then the normalized counting measure \(\nu_n\) on the zero set of \(q_n\) tends to the equilibrium measure \(\omega_S\) of \(S\) (for the logarithmic potential) in the weak\(^*\) topology, as \(n\) tends to \(\infty\). This article deals with a variety of similar statements from a point of view of orthogonality relations for polynomials, investigating the case of classical orthogonality, non-Hermitian orthogonality, orthogonality in rational approximation of Markov functions, and its non-Hermitian variant. The paper opens with a survey of basic concepts from potential theory that non-experts will find useful.