发表机构
Universitat de Barcelona; Centre de Recerca Matemàtica (CRM)（巴塞罗那大学; 数学研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明在球和非退化三角形区域上，$L^2$空间不存在指数的Riesz基，并推广到具有光滑边界的凸集，方法是通过构造平稳系统与投影比较族导出矛盾。
AI 中文摘要
我们证明当$\Omega$是$\R^n$（$n\geq2$）中的球或$\R^2$中的非退化三角形时，$L^2(\Omega)$不存在指数的Riesz基。球的结果推广到$\R^n$（$n\geq2$）中具有$C^2$边界的每个非空有界凸开集。从一个假设的Riesz基出发，我们使用其频率集平移的Beurling弱极限来构造一个平稳系统。在相关的Hilbert空间上，我们定义谱截断投影和由合成算子获得的正交投影的比较族。相应投影之间的距离一致地由一个严格小于1的常数界定。比较族在算子范数下是连续的，而$\Omega$的边界几何产生截断投影的两个强极限，其值域严格嵌套。这两个极限投影与同一比较投影的距离都将小于1，这是不可能的。
英文摘要
We prove that $L^2(Ω)$ admits no Riesz basis of exponentials when $Ω$ is a ball in $\R^n$, $n\geq2$, or a nondegenerate triangle in $\R^2$. The ball result extends to every nonempty bounded convex open set with $C^2$ boundary in $\R^n$, $n\geq2$. Starting from a hypothetical Riesz basis, we use Beurling weak limits of translates of its frequency set to construct a stationary system. On the associated Hilbert space, we define spectral cutoff projections and a comparison family of orthogonal projections obtained from the synthesis operator. The distance between corresponding projections is uniformly bounded by a constant strictly smaller than one. The comparison family is continuous in operator norm, whereas the boundary geometry of $Ω$ produces two strong limits of the cutoff projections with strictly nested ranges. Both limiting projections would then be at distance less than one from the same comparison projection, which is impossible.
CommentsIn this version we add a section that proves that any convex set has a complete minimal system of exponentials
