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

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.The authors would like to thank A. Sola for useful discussions, and the National Science Foundation for their support of the SEAM conference in 2016, where some of this work was carried out. Beneteau and Khavinson are grateful to the Centre de Recherches Mathematiques in Montreal for hosting them during the spring semester of 2016. Khavinson acknowledges the support of grant #513381 from the Simons Foundation. Liaw's work is supported by the National Science Foundation DMS-1802682. Seco acknowledges support from Ministerio de Economía y Competitividad Project MTM2014-51824-P and from Generalitat de Catalunya Project 2014SGR289

Remarks on Inner Functions and Optimal Approximants

We discuss the concept of inner function in reproducing kernelHilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to 1/f , where f is a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modiûed to produce inner functions

Analytic continuation of Dirichlet series

The questions considered in this paper arose from the study [KS] of I. Fredholm's (insufficient) proof that the gap series S08 an ?n2 (where 0 < |a| < 1) is nowhere continuable across {|?| = 1}. The interest of Fredholm's method ([F],[ML]) is not so much its efficacy in proving gap theorems (indeed, much more general results can be got by other means, cf. the Fabry gap theorem in [Di]) as in the connection it made between certain special gap series and partial differential equations (...)