Asymptotics for Hessenberg Matrices for the Bergman Shift Operator on Jordan Regions

作者：Saff Edward B^*; Stylianopoulos Nikos

来源：Complex Analysis and Operator Theory, 2014, 8(1): 1-24.

DOI：10.1007/s11785-012-0252-8

• Bergman orthogonal polynomials
• Faber polynomials
• Bergman shift operator
• Toeplitz matrix
• Strong asymptotics
• Conformal mapping

Let be a bounded Jordan domain in the complex plane. The Bergman polynomials of are the orthonormal polynomials with respect to the area measure over . They are uniquely defined by the entries of an infinite upper Hessenberg matrix . This matrix represents the Bergman shift operator of . The main purpose of the paper is to describe and analyze a close relation between and the Toeplitz matrix with symbol the normalized conformal map of the exterior of the unit circle onto the complement of . Our results are based on the strong asymptotics of . As an application, we describe and analyze an algorithm for recovering the shape of from its area moments.

• 出版日期2014-1