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arXiv · 2502.11805

Empirical plunge profiles of time-frequency localization operators

Simon Halvdansson
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Abstract

For time-frequency localization operators, related to the short-time Fourier transform, with symbol $R\Omega$, we work out the exact large $R$ eigenvalue behavior for rotationally invariant $\Omega$ and conjecture that the same relation holds for all scaled symbols $R \Omega$ as long as the window is the standard Gaussian. Specifically, we conjecture that the $k$-th eigenvalue of the localization operator with symbol $R\Omega$ converges to $\frac{1}{2}\operatorname{erfc}\big( \sqrt{2\pi}\frac{k-R^2|\Omega|}{R|\partial \Omega|} \big)$ as $R \to \infty$. To support the conjecture, we compute the eigenvalues of discrete frame multipliers with various symbols using LTFAT and find that they agree with the behavior of the conjecture to a large degree.

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Simon Halvdansson. 2025-02-17. Empirical plunge profiles of time-frequency localization operators. https://doi.org/10.1016/j.acha.2025.101825

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