2025/07/08 by Xinyu Chen, Bochen Liu, Chen, Xinyu +1 · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Mathematical Analysis and Transform Methods
paper · pdf · doi:10.48550/arxiv.2507.05777
openalex publication_date 2025/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ℝd near the north pole cannot have a frame spectrum near the xd-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 ℝd 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.