每个无限紧阿贝尔群上的递减重排傅里叶级数的发散性
Divergence of decreasing rearranged Fourier series on every infinite compact abelian group
浏览论文内容
中文总结 AI 辅助
该研究证明每个无限紧豪斯多夫阿贝尔群上存在复值连续函数，其递减重排傅里叶级数几乎处处发散，这类函数构成稠密Gδ集，通过结合相关引理与构造完成证明。
中文摘要 AI 辅助
在每个无限紧豪斯多夫阿贝尔群上，存在一个复值连续函数，其按幅度递减阈值选取系数的傅里叶和几乎处处无界。实际上，这类函数在带一致范数的连续函数空间中构成稠密的Gδ集。此外，可选取傅里叶支撑包含于对偶群任意指定无限子群且避开任意指定有限频率集的发散示例，且不要求群具有可度量化或零维性。证明结合了Körner的有限圆与沃尔什引理，以及奇循环群乘积上的有限构造；对偶群的三分法、精确商传递和循环采样为所考虑的每个群提供了有限示例。
英文摘要
On every infinite compact Hausdorff abelian group there is a complex-valued continuous function whose Fourier sums, taken over coefficients above a decreasing magnitude threshold, are unbounded almost everywhere. In fact, such functions form a dense \(G_δ\) in the space of continuous functions with its uniform norm. Moreover, a divergent example may be chosen with Fourier support contained in any prescribed infinite subgroup of the dual, while avoiding any prescribed finite set of frequencies. No metrizability or zero-dimensionality is assumed. The proof combines Körner's finite circle and Walsh lemmas with a finite construction on products of odd cyclic groups. A trichotomy for the dual group, exact quotient transfer, and cyclic sampling give finite examples on every group under consideration.
发表机构
- Aalborg University（奥尔堡大学）
机构由 AI 辅助整理，请以论文原文为准。
