Constructive solutions to Pólya–Schur problems

The general Pólya–Schur problem is to characterize linear operators on the space of univariate polynomials that preserve stability, where a polynomial is stable with respect to a region Ω in the complex plane if it has no zeros in Ω. Stable preserving operators have proven to be important in a variety of applications ranging from statistical mechanics to combinatorics, and variants of Pólya–Schur problems involving analytic functions are important in applications to signal processing. We present a structure theorem that bridges polynomial and analytic Pólya–Schur problems, providing constructive characterizations of stable-preserving operators for a general class of domains Ω. The structure theorem facilitates the solution of open Pólya–Schur problems in the classical setting, and provides constructive characterizations of stable preserving operators in cases where previously known characterizations are non-constructive. In the analytic setting, the structure theorem enables the explicit characterization of minimum-phase preserving operators on the half-line, a problem of importance in geophysical signal processing.