Asymptotics for Hessenberg matrices for the Bergman shift operator on\n Jordan regions

2012/05/18 by Edward B. Saff, Saff, Edward B., Nikos Stylianopoulos +1
Mathematics · #30C30 #30C50 #30C62 #41A10 #45Q05 (Secondary) #47B35 (Primary) 30C10 #Advanced Topics in Algebra #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical functions and polynomials #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1205.4183

openalex publication_date 2012/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a bounded Jordan domain in the complex plane and consider the\ninfinite upper Hessenberg matrix M associated with the Bergman orthogonal\npolynomials of G. This matrix represents the Bergman shift operator of G. The\nmain purpose of the paper is to describe and analyze a close relation between M\nand the Toeplitz matrix with symbol the normalized conformal map of the\nexterior of the unit circle onto the complement of the closure of G. Our\nresults are based on the strong asymptotics of the Bergman polynomials. As an\napplication, we describe and analyze an algorithm for recovering the shape of G\nfrom its area moments.\n

Citations

• Fine asymptotics for Bergman polynomials over domains with corners
• Strong asymptotics for Bergman polynomials over domains with corners and\n applications

Related

• The Hessenberg matrix and the Riemann mapping
• Bergman orthogonal polynomials and the Grunsky matrix
• Strong asymptotics for Bergman polynomials over domains with corners and\n applications
• Strong asymptotics for Bergman polynomials over domains with corners
• Spectral asymptotics for a class of Toeplitz operators on the Bergman space