On accumulated spectrograms

Abstract:

We study the eigenvalues and eigenfunctions of the time-frequency localization operator $H_\Omega$ on a domain $\Omega$ of the time-frequency plane. The eigenfunctions are the appropriate prolate spheroidal functions for an arbitrary domain $\Omega \subseteq \mathbb {R}^{2d}$. Indeed, in analogy to the classical theory of Landau-Pollak-Slepian, the number of eigenvalues of $H_\Omega$ in $[1-\delta , 1]$ is equal to the measure of $\Omega$ up to an error term depending on the perimeter of the boundary of $\Omega$. 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 of unity of the given domain $\Omega$. We derive asymptotic, non-asymptotic, and weak-$L^2$ error estimates for the accumulated spectrogram. As a consequence the domain $\Omega$ can be approximated solely from the spectrograms of eigenfunctions without information about their phase.