Zero Distribution of Bergman Orthogonal Polynomials for Certain Planar Domains |
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Authors: | Levin Saff Stylianopoulos |
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Affiliation: | Department of Mathematics The Open University of Israel 16 Klausner Street Tel-Aviv 61392 Israel elile@oumail.openu.ac.il, IL Center for Constructive Approximation Department of Mathematics Vanderbilt University Nashville, TN 37240 USA esaff@math.vanderbilt.edu, US Department of Mathematics and Statistics University of Cyprus P.O. Box 20537 1678 Nicosia Cyprus nikos@ucy.ac.cy, CY
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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 ? 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 ? 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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