Time-frequency localisation operators - a geometric phase space approach: II. The use of dilations.

*(English)*Zbl 0701.42004In Part I [IEEE Trans. Inf. Theory IT 34, No.4, 605-612 (1988; Zbl 0672.42007)] the first author defined operators which localized both in time and in frequency (i.e., low-pass filters which also restrict to a finite time interval), by means of a phase-space technique borrowed from quantum physics. The advantage of the localization operators constructed in the above-mentioned paper was the simplicity of their eigenfunctions (Hermitian functions) and their eigenvalues (simple and explicit expressions involving incomplete gamma functions). The present paper again concerns operators localizing in time and in frequency simultaneously, but constructed using a different procedure. The resulting operators restrict to a finite time interval, and cut off lows as well as high frequencies, i.e. are band-pass filters rather than low- pass filters. Again it will be possible to write explicit expressions for the eigenvalues and eigenfunctions.