2024/04/14 by Erwin Miña‐Díaz, Miña-Díaz, Erwin, Aron Wennman +1
Materials Science · Mathematics · #30C10 (Primary) 30E15 #30E20 (Secondary) #42C05 #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Flame retardant materials and properties #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2404.09335
openalex publication_date 2024/04/14 · openalex created_date 2024/04/17 · openalex updated_date 2026/07/28
We study the asymptotic behavior of the Bergman orthogonal polynomials (pn)n=0∞ for a class of bounded simply connected domains D. The class is defined by the requirement that conformal maps φ of D onto the unit disk extend analytically across the boundary L of D, and that φ' has a finite number of zeros z1,…, zq on L. The boundary L is then piecewise analytic with corners at the zeros of φ'. A result of Stylianopoulos implies that a Carleman-type strong asymptotic formula for pn holds on the exterior domain ℂ∖D. We prove that the same formula remains valid across L∖\z1,…,zq\ and on a maximal open subset of D. As a consequence, the only boundary points that attract zeros of pn are the corners. This is in stark contrast to the case when φ fails to admit an analytic extension past L, since when this happens the zero counting measure of pn is known to approach the equilibrium measure for L along suitable subsequences.