On the zero-free polynomial approximation problem

2015/01/01 by Arthur A. Danielyan, Danielyan, Arthur A.
Mathematics · #Advanced Banach Space Theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical functions and polynomials #math.CV

paper · pdf · doi:10.48550/arxiv.1501.00247

arxiv created 2015/01/01 · openalex publication_date 2015/01/01 · arxiv updated 2015/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let E be a compact set in \mathbb C with connected complement, and let A(E) be the class of all complex continuous function on E that are analytic in the interior E⁰ of E. Let f ∈ A(E) be zero free on E⁰. By Mergelyan's theorem f can be uniformly approximated on E by polynomials, but is it possible to realize such approximation by polynomials that are zero-free on E? This natural question has been proposed by J. Andersson and P. Gauthier. So far it has been settled for some particular sets E. The present paper describes classes of functions for which zero free approximation is possible on an arbitrary E.

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