*(English)*Zbl 1076.30010

Let $\mu $ be a finite positive Borel measure with infinite compact support $S\subset \mathbb{R}$ and consider the monic orthogonal polynomials ${q}_{n}\left(x\right)={x}^{n}+\cdots $ satisfying

A known result states that if $S$ is regular with respect to the Dirichlet problem in $\overline{\u2102}\setminus S$ 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.

##### MSC:

30C15 | Zeros of polynomials, etc. (one complex variable) |

30E10 | Approximation in the complex domain |

30E20 | Integration, integrals of Cauchy type, etc. (one complex variable) |

31A15 | Potentials and capacity, harmonic measure, extremal length (two-dimensional) |

05E35 | Orthogonal polynomials (combinatorics) (MSC2000) |

42C05 | General theory of orthogonal functions and polynomials |