How large are the gaps in phase space?
Michael Speckbacher
TL;DR
The paper provides explicit, quantitative upper bounds on the size of phase-space gaps for sampling measures associated with the wavelet transform (via Laguerre-based wavelets) and the short-time Fourier transform (via Hermite windows). By leveraging elementary transform formulas, the authors derive bounds that depend on the sampling-constant ratio $\frac{A}{B}$ and are independent of the detailed structure of the measure, with clear results for Bergman spaces as corollaries. The approach relies on the pseudohyperbolic geometry of phase space and an auxiliary ratio function, yielding concrete constants and extending to both continuous and discrete sampling frameworks. These results offer precise criteria for the relative denseness of sampling measures and connect wavelet and STFT sampling to well-studied function spaces like Bergman spaces and Gabor frames. The work provides practical gap estimates that can inform the design of sampling patterns in time-frequency analysis and quantum-mechanical models related to hyperbolic Landau levels.
Abstract
Given a sampling measure for the wavelet transform (resp. the short-time Fourier transform) with the wavelet (resp. window) being chosen from the family of Laguerre (resp. Hermite) functions, we provide quantitative upper bounds on the radius of any ball that does not intersect the support of the measure. The estimates depend on the condition number, i.e., the ratio of the sampling constants, but are independent of the structure of the measure. Our proofs are completely elementary and rely on explicit formulas for the respective transforms.