Zero Distribution of Bergman Orthogonal Polynomials for Certain Planar Domains

Abstract

Abstract. Let G be a simply connected domain in the complex plane bounded by a closed Jordan curve L and let P n , n≥ 0 , be polynomials of respective degrees n=0,1,··· that are orthonormal in G with respect to the area measure (the so-called Bergman polynomials). Let ϕ be a conformal map of G onto the unit disk. We characterize, in terms of the asymptotic behavior of the zeros of P n 's, the case when ϕ has a singularity on L . To investigate the opposite case we consider a special class of lens-shaped domains G that are bounded by two orthogonal circular arcs. Utilizing the theory of logarithmic potentials with external fields, we show that the limiting distribution of the zeros of the P n 's for such lens domains is supported on a Jordan arc joining the two vertices of G . We determine this arc along with the distribution function.

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Levin, Saff & Stylianopoulos Zero Distribution of Bergman Orthogonal Polynomials for Certain Planar Domains . Constr. Approx. 19, 411–435 (2003). https://doi.org/10.1007/s00365-002-0519-9

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  • Key words. Bergman orthogonal polynomials, Zeros of polynomials, Weighted potentials, Equilibrium measure, Balayage measure. AMS Classification. 30C10, 30C15, 30C85, 31A05, 31A15.