Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations pp 399-412 | Cite as

# 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.

## Keywords

Non-negativity polynomials exponential functions trigonometric polynomials generalized Budan-Fourier sequence Kronecker’s approximation theorem Lipschitz continuity## Preview

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