Constructive solutions to Pólya-Schur problems

We present constructive solutions to the following Pólya-Schur problems concerning linear operators on the space of univariate polynomials: Given subsets $Ω_1$ and $Ω_2$ of the complex plane, determine operators that map all polynomials having no zeros in $Ω_1$ to polynomials having no zeros in $Ω_2$, 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 $Ω_1$ and $Ω_2$; and this class is shown to comprise all solutions when $Ω_1$ is bounded and $Ω_2$ has non-empty interior. The latter result encompasses a number of open problems and, moreover, gives explicit solutions in cases of circular domains $Ω_1=Ω_2$ 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.

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