Constructive solutions to Pólya–Schur problems

Abstract

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.