Revista Matemática Iberoamericana


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Published online first: 2019-02-19
DOI: 10.4171/rmi/1064

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

Catherine Bénéteau[1], Dmitry Khavinson[2], Constanze Liaw[3], Daniel Seco[4] and Brian Simanek[5]

(1) University of South Florida, Tampa, USA
(2) University of South Florida, Tampa, USA
(3) University of Delaware, Newark, USA
(4) Universidad Carlos III de Madrid, Leganés, Spain
(5) Baylor University, Waco, USA

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.

Keywords: Bergman spaces, Dirichlet spaces, cyclic functions, orthogonal polynomials

Bénéteau Catherine, Khavinson Dmitry, Liaw Constanze, Seco Daniel, Simanek Brian: Zeros of optimal polynomial approximants: Jacobi matrices and Jentzsch-type theorems. Rev. Mat. Iberoam. Electronically published on February 19, 2019. doi: 10.4171/rmi/1064 (to appear in print)