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Zero distributions via orthogonality. (English) Zbl 1076.30010

Let μ be a finite positive Borel measure with infinite compact support S and consider the monic orthogonal polynomials q n (x)=x n + satisfying

q n (t)t k dμ(t)=0,k=0,1,...,n-1·

A known result states that if S is regular with respect to the Dirichlet problem in ¯S and if μ is “sufficiently thick,” then the normalized counting measure ν n on the zero set of q n tends to the equilibrium measure ω S of S (for the logarithmic potential) in the weak * topology, as n tends to . 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.

30C15Zeros of polynomials, etc. (one complex variable)
30E10Approximation in the complex domain
30E20Integration, integrals of Cauchy type, etc. (one complex variable)
31A15Potentials and capacity, harmonic measure, extremal length (two-dimensional)
05E35Orthogonal polynomials (combinatorics) (MSC2000)
42C05General theory of orthogonal functions and polynomials