Phase space localized functions in the Dunkl setting

Abstract

Time–frequency analysis plays a central role in signal analysis, because signals that have highly concentrated time–frequency content are used in many applications. The uncertainty principle in Fourier analysis sets a limit to the possible simultaneous concentration of a function and its Dunkl transform. To localize signals in the time–frequency plane we use time–frequency localization operators in the Dunkl setting to measure their time–frequency content on some subset of finite measure. Then, using eigenfunctions of these operators, which are maximally time–frequency concentrated, we prove a characterization of functions that are time–frequency concentrated in the region of interest, and we obtain approximation inequalities for such functions using a finite linear combination of eigenfunctions.