Abstract

In this paper, we show that the surface measure on the boundary of a convex body of everywhere positive Gaussian curvature does not admit a Fourier frame. This answers a question proposed by Lev and provides the 1st example of a uniformly distributed measure supported on a set of Lebesgue measure zero that does not admit a Fourier frame. In contrast, we show that the surface measure on the boundary of a polytope always admits a Fourier frame. We also explore orthogonal bases and frames adopted to sets under consideration. More precisely, given a compact manifold $M$ without a boundary and $D \subset M$, we ask whether $L^2(D)$ possesses an orthogonal basis of eigenfunctions. The non-abelian nature of this problem, in general, puts it outside the realm of the previously explored questions about the existence of bases of characters for subsets of locally compact abelian groups. This paper is dedicated to Alexander Olevskii on the occasion of his birthday. Olevskii’s mathematical depth and personal kindness serve as a major source of inspiration for us and many others in the field of mathematics.

中文翻译：

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用于地面测量的傅立叶框架

摘要

在本文中，我们证明了处处为正高斯曲率的凸体边界上的表面测度不允许傅立叶框架。这回答了 Lev 提出的问题，并提供了在一组 Lebesgue 测度零上支持的均匀分布测度的第一个示例，该测度零不允许傅立叶框架。相反，我们表明多面体边界上的表面测量总是允许傅立叶框架。我们还探索了所考虑的集合采用的正交基和框架。更准确地说，给定一个没有边界的紧流形 $M$ 和 $D\subset M$，我们问 $L^2(D)$ 是否具有特征函数的正交基。这个问题的非阿贝尔性质，一般来说，将其置于先前探索的关于局部紧致阿贝尔群子集的字符基是否存在的问题之外。这篇论文是献给亚历山大·奥列夫斯基生日之际的。Olevskii 的数学深度和个人友善是我们和数学领域其他许多人的主要灵感来源。