Sharp rates of convergence for accumulated spectrograms

Abstract

Abstract. We investigate an inverse problem in time-frequency localization: the approximation of the symbol of a time-frequency localization operator from partial spectral information by the method of accumulated spectrograms (the sum of the spectrograms corresponding to large eigenvalues). We derive a sharp bound for the rate of convergence of the accumulated spectrogram, improving on recent results.

On Accumulated Cohen’s Class Distributions and Mixed-State Localization Operators

“…The main object of this paper is an in-depth treatment of the mixed-state localization operators from [29] and their associated time-frequency distributions from the perspective developed in [2,3], and to describe them we first recall some facts about quantum harmonic analysis [27,36]. Concretely, the convolution between two trace class operators and the convolution between a function and a trace class operator.…”

“…. Quantum harmonic analysis seems to provide the natural setting for the investigations of eigenvalues and eigenvectors of (mixed-state) localization operators as in this setup many of the proofs in [2,3,14] become natural statements about convolutions between operators. An important aspect of this paper is that one can reformulate the results of [2] in terms of quantum harmonic analysis which then allows us to formulate their results for mixed-state localization operators.…”

“…Let us briefly present our results: The first result is that the eigenvalues of a mixed-state localization operator has the same asymptotic behaviour as the one for localization operators [14,32], see theorem 4.4. This is a prerequisite for generalizing the results [2,3]. A key fact is that the approach in [2,3] is only feasible in the case of rank-one operators.…”

“…This is a prerequisite for generalizing the results [2,3]. A key fact is that the approach in [2,3] is only feasible in the case of rank-one operators. For a general density operator one has to develop a different strategy.…”

“…Our main results are the extension of the theorems in [2,3] on the accumulated spectrogram to accumulated Cohen's class distributions. Our proofs are non-trivial adaptations of the ones in [2,3] and we have tried to emphasize the modifications required by the mixed-state setting.…”