Analytic closure of sets of real hyperbolic polynomials with separated roots

Abstract

We characterize closures (in the topology of local uniform convergence on \({\mathbb {C}}\)) of certain classes of hyperbolic polynomials (real polynomials with no complex roots) in terms of restricted versions of the classical Laguerre–Polya class. Our results are motivated by recent considerations of Fisk and others of the interlacing-preserving linear operators on hyperbolic polynomials. We discuss closures of classes of hyperbolic polynomials with mesh (minimal root separation) bounded from below and of hyperbolic polynomials with logarithmic mesh (minimal ratio of successive roots) bounded from below. We also provide some examples and applications.