2011/07/30 by Peter C. Gibson, Michael P. Lamoureux, Gibson, Peter C. +1
Mathematics · #30 #47 #Complex Variables (math.CV) #FOS: Mathematics #math.CV #msc:30 #msc:47
paper · pdf · doi:10.48550/arxiv.1108.0043
Final version, incorporating referees' comments, 15 pages
arxiv created 2015/09/08 · arxiv updated 2015/09/09
We present constructive solutions to the following Pólya-Schur problems concerning linear operators on the space of univariate polynomials: Given subsets Ω1 and Ω2 of the complex plane, determine operators that map all polynomials having no zeros in Ω1 to polynomials having no zeros in Ω2, 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 Ω1 and Ω2; and this class is shown to comprise all solutions when Ω1 is bounded and Ω2 has non-empty interior. The latter result encompasses a number of open problems and, moreover, gives explicit solutions in cases of circular domains Ω1=Ω2 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.