The Hermite-Biehler method and functions generating the Pólya frequency sequences

Abstract

Under the Hermite-Biehler method we understand the approach to problems of stability, which exploits a deep relation between Hurwitz-stable functions and mappings of the upper half of the complex plane into itself (i.e. R-functions). This method dates back to works by Hermite, Biehler, Hurwitz; in the first half of the XXth century it was extended to entire functions by Grommer and Krein. Recent papers (from 1970 to the present date) highlighted another property related to Hurwitz-stability: the total nonnegativity of the corresponding Hurwitz matrix, that is nonnegativity of all its minors. Since each Hurwitz matrix is built from two Toeplitz matrices, this also involves the class PF of functions corresponding to totally nonnegative Toeplitz matrices (i.e. generating functions of Polya frequency sequences). The present work investigates connections between R-functions, PF -functions and the localization of zeros (e.g. Hurwitz stability). The first problem we deal with is to prove that zeros of one remarkable family of polynomials are interlacing. This family originates from the Jacobi tau method for the Sturm-Liouville eigenvalue problem, and the interlacing property guarantees that the spectra of approximations are real. At that, the Hurwitz stability comes from a specially composed differential equation, and then the Hermite-Biehler theorem implies the interlacing property for pairs of polynomials. The next topic is a complete description of functions generating the infinite totally nonnegative Hurwitz matrices. Our results exploit a connection between a factorization of the Hurwitz-type matrices and the expansion of R-functions into Stieltjes continued fractions. Further we study solutions to the equation zpR(zk ) = α with nonzero complex α, integer p, k and R ∈ PF by relating it to the class R. Such equations appear from manipulations akin to the Hermite-Biehler method when we prove that functions of the form ∑∞ n=0(±i)anz have simple zeros distinct in absolute value under a certain condition on the coefficients an ⩾ 0. Lastly, we study the generalized Nevanlinna classesN + < and their connection to PF -functions. The present work elaborates the criterion given by Krein and Langer and applies it to counting the number of zeros and poles of PF -functions.