Time-frequency partitions for the Gelfand triple $(S_0,L^2,S_0')$

We give a new characterization of the Gelfand triple of function spaces in $(S_0, L^2, S_0')$ by means of a family of time-frequency localization operators. The localization operators are defined by the short-time Fourier transform and determine the local time-frequency behavior, whereas the global time-frequency distribution is characterized by a sequence space norm. We also show that the alternative time-frequency localization method with the Weyl transform fails to yield a similar characterization of time-frequency distribution.