Gabor Transform Associated with the Dunkl–Bessel Transform and Spectrograms

Time–frequency (or space–phase) analysis plays a key role in signal analysis. In particular, signals that have a very concentrated time–frequency content are of great importance. However, the uncertainty principle sets a limitation to the possible simultaneous concentration of a function and its Dunkl–Bessel transform. For this purpose, we introduce and study a new transformation called Dunkl–Bessel Gabor transform. For this transformation, we define the Toeplitz-type (or time–frequency localization) operators, in order to localize signals on the time–frequency plane. We study these operators; in particular, we give criteria for their boundedness and Schatten class properties. Then, using the special class of concentration operators, which are compact and self-adjoint, we show that their eigenfunctions are maximally time–frequency-concentrated in the region of interest.