Fourier frames on smooth surfaces with nonvanishing Gaussian curvature

TL;DR

This paper proves that certain smooth surfaces with nonvanishing Gaussian curvature, including small caps and hemispheres, do not admit Fourier frames, extending previous results and resolving open questions in harmonic analysis.

Contribution

It generalizes prior work by showing that smooth surfaces with nonvanishing Gaussian curvature cannot have Fourier frames, including endpoint cases like hemispheres.

Findings

Small spherical caps lack Fourier frames.

Hemispheres do not admit Fourier frames.

Results extend to general smooth surfaces with nonvanishing Gaussian curvature.

Abstract

It is known that a small spherical cap (rigorously its surface measure) admits Fourier frames, while the whole sphere does not. In this paper, we prove more general results. Consequences indclude that a small spherical cap in $\mathbb{R}^d$ near the north pole cannot have a frame spectrum near the $x_d$-axis, and $S$ does not admit any Fourier frame if its interior contains a closed hemisphere. We also resolve the endpoint case, that is, a hemisphere does not admit any Fourier frame. This answers a question of Kolountzakis and Lai. Our results also hold on more general smooth surfaces with nonvanishing Gaussian curvature. In particular, any compact $(d-1)$-dimensional smooth submanifold immersed in $\mathbb{R}^d$ with nonvanishing Gaussian curvature does not admit any Fourier frame. This generalizes a previous result of Iosevich, Lai, Wyman and the second author on the boundary of convex bodies, as well as improves a recent result of Kolountzakis and Lai from tight frame to frame.