Constructive solutions to P\'olya-Schur problems

Abstract

We present constructive solutions to the following P\'olya-Schur problems concerning linear operators on the space of univariate polynomials: Given subsets $\Omega_1$ and $\Omega_2$ of the complex plane, determine operators that map all polynomials having no zeros in $\Omega_1$ to polynomials having no zeros in $\Omega_2$, or to the zero polynomial. We describe an explicit class consisting of rank 1 operators and product-composition operators that solve the stated problems for arbitrary $\Omega_1$ and $\Omega_2$; and this class is shown to comprise all solutions when $\Omega_1$ is bounded and $\Omega_2$ has non-empty interior. The latter result encompasses a number of open problems and, moreover, gives explicit solutions in cases of circular domains $\Omega_1=\Omega_2$ where existing characterizations are non-constructive. The paper also treats problems stemming from digital signal processing that are analogous to P\'olya-Schur problems. Specifically, we describe all bounded linear operators on Hardy space that preserve the class of outer functions, as well as those that preserve shifted outer functions.