The Winding Number at the Critical H\"older Exponent 1/3: Failure of Universal Fourier Summation

Abstract

The degree (or winding number) of a sufficiently regular map $f:\mathbb{T} \to \mathbb {S^1}$ is given in terms of its Fourier coefficients by $$ \operatorname{deg} f = \sum_{n\in\mathbb{Z}} n\,|\widehat{f}(n)|^2. $$ At lower regularity the series may diverge, but, as shown by Kahane, when $f$ is $\alpha$-H\"older continuous with $\alpha>1/3$, then the degree can be recovered by a universal linear summation process. We show that no summation process satisfying Brezis's natural axioms can recover the degree universally for $\alpha=1/3$, thereby resolving an open problem by Brezis.