AI 中文总结
研究在临界赫尔德指数1/3时通用傅里叶求和能否恢复映射次数的问题，通过证明不存在满足布雷齐斯自然公理的求和过程能普遍恢复次数，解决了布雷齐斯提出的开放问题。
AI 中文摘要
对于足够正则的映射$f:\mathbb{T} \to \mathbb {S^1}$，其次数（或缠绕数）由其傅里叶系数通过公式$\operatorname{deg} f = \sum_{n\in\mathbb{Z}} n\,|\widehat{f}(n)|^2$给出。在较低正则性下该级数可能发散，但如卡哈内所示，当$f$是$\alpha$-赫尔德连续且$\alpha>1/3$时，次数可通过通用线性求和过程恢复。我们证明，对于$\alpha = 1/3$，没有满足布雷齐斯自然公理的求和过程能普遍恢复次数，从而解决了布雷齐斯的一个开放问题。
英文摘要
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 $α$-Hölder continuous with $α>1/3$, 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 $α=1/3$, thereby resolving an open problem by Brezis.
