Time-frequency localization operators: A geometric phase space approach

We define a set of operators which localize in both time and frequency. Tltese operators are similar to but different from the low-pass time-liting operators, the singular functions of which are the prolate spheroidal wave functions. Our construction differs from the usual ap proach in that we treat the time-frequency plane as one geometric whole (phase space) rather than as two separate spaces. For disk-shaped or ellipse-shaped domains in the time-frequency plane, the associated localization operators are remarkably simple. ”heir eigenfunctions are Hermite functions, and the corresponding eigenvalues are given by simple explicit formulas involving the incomplete gamma functions.