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

publication date

  • February 2019

start page

  • 607

end page

  • 642

issue

  • 2

volume

  • 35

International Standard Serial Number (ISSN)

  • 0213-2230

Electronic International Standard Serial Number (EISSN)

  • 2235-0616

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.

keywords

  • bergman spaces; dirichlet spaces; cyclic functions; orthogonal polynomials