JournalsrmiVol. 35, No. 2pp. 607–642

Zeros of optimal polynomial approximants: Jacobi matrices and Jentzsch-type theorems

  • Catherine Bénéteau

    University of South Florida, Tampa, USA

  • Dmitry Khavinson

    University of South Florida, Tampa, USA

  • Constanze Liaw

    University of Delaware, Newark, USA

  • Daniel Seco

    Universidad Carlos III de Madrid, Leganés, Spain

  • Brian Simanek

    Baylor University, Waco, USA
Zeros of optimal polynomial approximants: Jacobi matrices and Jentzsch-type theorems cover
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Abstract

We study the structure of the zeros of optimal polynomial approximants to reciprocals of functions in Hilbert spaces of analytic functions in the unit disk. In many instances, we find the minimum possible modulus of occurring zeros via a nonlinear extremal problem associated with norms of Jacobi matrices. We examine global properties of these zeros and prove Jentzsch-type theorems describing where they accumulate. As a consequence, we obtain detailed information regarding zeros of reproducing kernels in weighted spaces of analytic functions.

Cite this article

Catherine Bénéteau, Dmitry Khavinson, Constanze Liaw, Daniel Seco, Brian Simanek, Zeros of optimal polynomial approximants: Jacobi matrices and Jentzsch-type theorems. Rev. Mat. Iberoam. 35 (2019), no. 2, pp. 607–642

DOI 10.4171/RMI/1064