Nonharmonic multivariate Fourier transforms and matrices: Condition numbers and hyperplane geometry

• Published in: Applied and computational harmonic analysis Vol. 79; p. 101791
• Main Author: Li, Weilin
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.10.2025
• Subjects: Fourier matrix; Fourier transform; Nonharmonic; Singular values; Sparsity; Trigonometric interpolation; Trigonometric interpolation; 42A15; Sparsity; 15A12; 65F22; Fourier matrix; Singular values; Fourier transform; Nonharmonic; 15A60; 42A05
• ISSN: 1063-5203
• DOI: 10.1016/j.acha.2025.101791

Consider an operator that takes the Fourier transform of a discrete measure supported in X⊆[−12,12)d and restricts it to a compact Ω⊆Rd. We provide lower bounds for its smallest singular value when Ω is either a closed ball of radius m or closed cube of side length 2m, and under different types of geometric assumptions on X. We first show that if distances between points in X are lower bounded by a δ that is allowed to be arbitrarily small, then the smallest singular value is at least Cmd/2(mδ)λ−1, where λ is the maximum number of elements in X contained within any ball or cube of an explicitly given radius. This estimate communicates a localization effect of the Fourier transform. While it is sharp, the smallest singular value behaves better than expected for many X, including when we dilate a generic set by parameter δ. We next show that if there is a η such that, for each x∈X, the set X∖{x} locally consists of at most r hyperplanes whose distances to x are at least η, then the smallest singular value is at least Cmd/2(mη)r. For dilations of a generic set by δ, the lower bound becomes Cmd/2(mδ)⌈(λ−1)/d⌉. The appearance of a 1/d factor in the exponent indicates that compared to worst case scenarios, the condition number of nonharmonic Fourier transforms is better than expected for typical sets and improve with higher dimensionality.