Gabor Frames of Totally Positive Functions: A Complete Characterization

What the math gives to ML

The paper gives an exact density criterion for time-frequency systems generated by continuous integrable totally positive windows: the Gabor atoms form a complete stable frame precisely when the time-frequency cell satisfies \(\alpha\beta<1\). This can transfer to neural signal-processing front-ends as a principled constraint on temporal hop size and frequency spacing, replacing heuristic filterbank subsampling. The most practical architecture is a differentiable, oversampled Gabor layer with a fixed or constrained totally positive window and a parameterization that guarantees \(\alpha\beta<1\), providing a coverage and stability prior while preserving end-to-end learning.