The Winding Number at the Critical Hölder 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 $$\mathrm{deg}f=\sum _{n\in \mathbb{Z}}n|\hat{f}(n){|}^2.$$ At lower regularity the series may diverge, but, as shown by Kahane, when $f$ is $\alpha$-Hölder 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.