Fourier Frames for Surface-Carried Measures

Alex Iosevich, Chun Kit Lai, Bochen Liu, Emmett Wyman

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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 ⊂ M, we ask whether L2(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.

Original languageEnglish
Pages (from-to)1644-1665
Number of pages22
JournalInternational Mathematics Research Notices
Volume2022
Issue number3
DOIs
StatePublished - 1 Feb 2022
Externally publishedYes

Bibliographical note

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© The Author(s) 2020.

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