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

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.

A Characterization of Dirichlet-inner Functions

“…This may yield an additional way to prove the result in this article in a more general environment. The dichotomic situation between spaces with or without the expansive shift property is also present in previous work ( [7]). In spaces where the shift is contractive, an analogue proof to that of our theorem will show that contractive divisors are inner, although in the most classical case this is already well known.…”

Remarks on Inner Functions and Optimal Approximants

“…To prove (a), notice that n∈N P n f is dense in [f ]. Since H is a Hilbert space, the orthogonal projection of 1 onto P n f , p * n f , must converge to the orthogonal projection of 1 onto [f ], that is, h. (This fact was already noticed by the authors in [4]. )…”

“…Optimal approximants were further studied in a subsequent series of papers [3,4], and it seems worthwhile to isolate additional properties of functions G that satisfy the orthogonality relations…”

Hyponormal Toeplitz operators with non-harmonic algebraic symbol

“…Remark. Notice that the extremal problem posed in (3) is similar to that considered in [2], but is not identical, so the results of that paper cannot be directly applied. Corollary 2.3.…”