# Fourier frames for surface-carried measures

## TL;DR

This paper investigates the existence of Fourier frames for surface measures on convex bodies and polytopes, revealing that smooth convex boundaries lack Fourier frames while polytopes do admit them, and explores related orthogonal basis questions.

## Contribution

It provides the first example of a Lebesgue measure-zero set without a Fourier frame and characterizes when surface measures on polytopes admit Fourier frames.

## Key findings

- Surface measure on convex bodies with positive Gaussian curvature does not admit a Fourier frame.
- Surface measure on polytopes always admits a Fourier frame.
- The paper explores orthogonal bases for functions on manifolds and subsets, extending beyond abelian group settings.

## 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 first 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.