Sharp sign uncertainty for trigonometric polynomials

Abstract

We study sign uncertainty principles for trigonometric polynomials of prescribed degree $N$ with respect to a symmetric Borel measure $μ$ on the unit circle $\mathbb{R}/\mathbb{Z}$. For each such measure, we determine the smallest radius of the last sign change for trigonometric polynomials with non-positive $μ$-integral. We further extend these results to polar measures on higher-dimensional spheres $\mathbb{S}^d$, showing that the extremal problem reduces to the one-dimensional case via the polar part of the measure, and we establish a polynomial analogue on $[0,1]$ using orthogonal polynomials on the real line.