Phase space localization operators: asymptotics of singular values

The Weyl correspondence is a convenient and practical way to define a broad class of time-frequency localization operators. Given a region Ω in the time-frequency (or phase) plane R² , and a distribution P(ξ, x) in the Cohen´s (1989) class, the Weyl correspondence defines an operator L_Ω^P on L²(R) which essentially localizes the signal in the region Ω according to the distribution P. Different choices of P allow different interpretations of phase-space localization. Empirically, such localization operators have the following singular value structure: the singular values range between 0 and 1, and when arranged in decreasing order, a finite number of them are close to 1, while the rest fall toward 0 asymptotically. We present a result on the asymptotic structure of the singular values of such phase-space localization operators