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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.

MSC:
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