Linear preservers and entire functions with restricted zero loci

Abstract

Let T : R [x] → R [x] be a linear operator such that T[ xk] = gammakxk for all k = 0, 1, 2, ..., where gamma k ∈ R . The real sequence gkinfinity k=0 is called a multiplier sequence if for any p ∈ R [x], having only real zeros, T[ p] also has only real zeros. A characterization of all multiplier sequences that can be interpolated by rational functions is given. This partially solves a problem of G. Csordas and T. Craven, who asked for a characterization of all the meromorphic functions, Y(k), such that Yk infinityk=0 is a multiplier sequence. An eight-year-old conjecture of I. Krasikov is proved. Several discrete analogues of classical inequalities for polynomials with only real zeros are obtained, along with results which allow extensions to a class of transcendental entire functions in the Laguerre-Polya class. A study of finite difference operators which preserve reality of zeros is initiated, and new results are proved. Composition theorems and inequalities for polynomials having their zeros in a sector are obtained. These are analogs of classical results by Polya, Schur, and Turan. In addition, a result of Obreschkoff is used to show that the Jensen polynomials related to the Riemann xi-function have only real zeros up to degree 1017. Sufficient conditions are established for a linear transformation to map polynomials having zeros only in a sector to polynomials of the same type, and some multivariate extensions of these results are presented. A complete characterization is given for linear operators which preserve closed ("strict") half-plane stability in the univariate Weyl algebra. These results provide new information about a general stability problem posed formally by G. Csordas and T. Craven. In his 2011 AMS Bulletin article, D. G. Wagner describes recent activity in multivariate stable polynomial theory as "exciting work---elementary but subtle, and with spectacular consequences." Wagner points out that many of the recent advancements in the theory of multivariate stable polynomials are due to the pioneering work of J. Borcea and P. Branden. These results play an important role in the investigation of linear stability preservers in this dissertation. Several different approaches to characterizing linear transformations which map polynomials having zeros only in one region of the complex plane to polynomials of the same type are explored. In addition, an open problem of S. Fisk is solved, and several partial results pertaining to open problems from the 2007 AIM workshop "Polya-Schur-Lax problems: hyperbolicity and stability preservers" are obtained.