A New Family of Nonnegative Sine Polynomials

TL;DR

This paper introduces a new family of sine polynomials that are nonnegative over [0, π], characterizes all nonnegative degree 3 sine polynomials and degree 2 cosine polynomials, and corrects previous typos.

Contribution

It presents a novel family of nonnegative sine polynomials and complete characterizations for specific degrees, advancing understanding of polynomial nonnegativity.

Findings

Introduces a new family of nonnegative sine polynomials

Characterizes all nonnegative degree 3 sine polynomials

Characterizes all nonnegative degree 2 cosine polynomials

Abstract

We present a new family of sine polynomials that are nonnegative for all $x$ in $[0,\pi]$. We also characterize all nonnegative sine polynomials of degree 3 and all nonnegative cosine polynomials of degree 2. In the latest version, typos in (1.4) and (1.6) are corrected (with >= replaced by <=).