Empirical plunge profiles of time-frequency localization operators

TL;DR

This paper investigates the asymptotic behavior of eigenvalues of time-frequency localization operators with scaled symbols, providing exact results for rotationally invariant cases and supporting the conjecture through numerical experiments.

Contribution

It derives the large-scale eigenvalue asymptotics for a class of localization operators and conjectures their general form, supported by numerical evidence.

Findings

Eigenvalues follow a specific error function profile as scale increases.

Numerical computations align with the conjectured asymptotic behavior.

Exact asymptotics are established for rotationally invariant symbols.

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.