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On accumulated spectrograms

We study the eigenvalues and eigenfunctions of the time-frequency localization operator $H_Ω$ on a domain $Ω$ of the time-frequency plane. The eigenfunctions are the appropriate prolate spheroidal functions for an arbitrary domain $Ω$. Indeed, in analogy to the classical theory of Landau-Slepian-Pollak, the number of eigenvalues of $H_Ω$ in $[1-δ, 1]$ is equal to the measure of $Ω$ up to an error term depending on the perimeter of the boundary of $Ω$. Our main results show that the spectrograms of the eigenfunctions corresponding to the large eigenvalues (which we call the accumulated spectrogram) form an approximate partition ofunity of the given domain $Ω$. We derive both asymptotic, non-asymptotic, and weak $L^2$ error estimates for the accumulated spectrogram. As a consequence the domain $Ω$ can be approximated solely from the spectrograms of eigenfunctions without information about their phase.