ON ACCUMULATED SPECTROGRAMS

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 subset of 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.