Nonnegative Trigonometric Polynomials and Sturms Theorem

TL;DR

This paper introduces a new algorithm based on Sturm's Theorem that can rigorously verify the nonnegativity of all types of trigonometric polynomials, including those with both sine and cosine terms.

Contribution

It extends previous algorithms to handle general trigonometric polynomials involving both sine and cosine terms, using classical Sturm's Theorem.

Findings

Algorithm successfully verifies nonnegativity of general trigonometric polynomials.

Enhances previous methods by providing a comprehensive solution.

Supports ongoing research in nonnegative trigonometric polynomial analysis.

Abstract

In an earlier article [3], we presented an algorithm that can be used to rigorously check whether a specific cosine or sine polynomial is nonnegative in a given interval or not. The algorithm proves to be an indispensable tool in establishing some recent results on nonnegative trigonometric polynomials. See, for example, [2], [4] and [5]. It continues to play an essential role in several ongoing projects. The algorithm, however, cannot handle general trigonometric polynomials that involve both cosine and sine terms. Some ad hoc methods to deal with such polynomials have been suggested in [3], but none are, in general, satisfactory. This note supplements [3] by presenting an algorithm applicable to all general trigonometric polynomials. It is based on the classical Sturm Theorem, just like the earlier algorithm. A couple of the references in [3] are also updated.