Non-negativity Analysis for Exponential-Polynomial-Trigonometric Functions on [0,∞)

Abstract

This note concerns the class of functions that are solutions of homogeneous linear differential equations with constant real coefficients. This class, which is ubiquitous in the mathematical sciences, is denoted throughout the paper by EPT, and members of it can be written in the form ${\sum\limits^{d}_{i=0}}{q_i}(t)e^{\lambda_{i}t}{\rm cos}({\theta_i}{t}+{\tau_i}),$ where the qi are real polynomials, and λi, θi and τi are real numbers. The subclass of these functions, for which all the θi are zero, is denoted by EP. In this paper, we address the characterization of those members of EPT that are non-negative on the half-line [0, ∞). We present necessary conditions, some of which are known, and a new sufficient condition, and describe methods for the verification of this sufficient condition. The main idea is to represent an EPT function as the product of a row vector of EP functions, and a column vector of multivariate polynomials with unimodular exponential functions ${\rm e}^{{i\theta}_{k}{t}},\,k=1,\,2,\,\ldots,m,$, as arguments, where $\{{\theta_k}\,:k=1,2,\ldots,m\}$ is a set of real numbers that is linearly independent over the set of rational numbers Q. From this we deduce necessary conditions for an EPT function to be non-negative on an unbounded subinterval of [0, ∞). The completion of the analysis is reliant on a generalized Budan-Fourier sequence technique, devised by the authors, to examine the non-negativity of the function on the complementary interval.

Mathematics Subject Classification (2000). Primary 34H05; Secondary 33B10.