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Zbl 1076.30010
Baratchart, Laurent; Küstner, Reinhold; Totik, Vilmos
Zero distributions via orthogonality. (English)
[J] Ann. Inst. Fourier 55, No. 5, 1455-1499 (2005). ISSN 0373-0956; ISSN 1777-5310/e

Let $\mu$ be a finite positive Borel measure with infinite compact support $S\subset\Bbb R$ 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 $\overline{\Bbb{C}}\setminus S$ 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.
[Vania Mascioni (Muncie)]

MSC 2000:: *30C15 Zeros of polynomials, etc. (one complex variable)
30E10 Approximation in the complex domain
30E20 Integration (one complex variable)
31A15 Potentials, etc. (two-dimensional)
05E35 Orthogonal polynomials (combinatorics)
42C05 General theory of orthogonal functions and polynomials

Keywords: orthogonal polynomials; zero distribution; logarithmic potential; rational approximation

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