Empirical plunge profiles of time-frequency localization operators
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909932626903040 |
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| author | Halvdansson, Simon |
| author_facet | Halvdansson, Simon |
| contents | For time-frequency localization operators, related to the short-time Fourier transform, with symbol $RΩ$, we work out the exact large $R$ eigenvalue behavior for rotationally invariant $Ω$ and conjecture that the same relation holds for all scaled symbols $R Ω$ as long as the window is the standard Gaussian. Specifically, we conjecture that the $k$-th eigenvalue of the localization operator with symbol $RΩ$ converges to $\frac{1}{2}\operatorname{erfc}\big( \sqrt{2π}\frac{k-R^2|Ω|}{R|\partial Ω|} \big)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_11805 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Empirical plunge profiles of time-frequency localization operators Halvdansson, Simon Functional Analysis Classical Analysis and ODEs For time-frequency localization operators, related to the short-time Fourier transform, with symbol $RΩ$, we work out the exact large $R$ eigenvalue behavior for rotationally invariant $Ω$ and conjecture that the same relation holds for all scaled symbols $R Ω$ as long as the window is the standard Gaussian. Specifically, we conjecture that the $k$-th eigenvalue of the localization operator with symbol $RΩ$ converges to $\frac{1}{2}\operatorname{erfc}\big( \sqrt{2π}\frac{k-R^2|Ω|}{R|\partial Ω|} \big)$ 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. |
| title | Empirical plunge profiles of time-frequency localization operators |
| topic | Functional Analysis Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2502.11805 |