Constructive solutions to Pólya-Schur problems

Abstract

We present constructive solutions to the following Pólya-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 Pólya-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.

Final version, incorporating referees' comments, 15 pages