Time-frequency localization operators: A geometric phase space approach.

We define a set of operators which localize in both time and frequency. These operators are similar to but different from the low-pass, time-limiting operators, the singular functions of which are the prolate spheroidal wave functions. Our construction differs from the usual approach 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. Their eigenfunctions are Hermite functions, and the corresponding eigenvalues are given by simple, explicit formulas involving the incomplete gamma-functions.