Construction of warped time-frequency representations on nonuniform frequency scales, Part II: Integral transforms, function spaces, atomic decompositions and Banach frames

Abstract

We present a novel family of continuous linear time-frequency transforms adapted to a multitude of (nonlinear) frequency scales. Similar to classical time-frequency or time-scale representations, the representation coefficients are obtained as inner products with the elements of a continuously indexed family of time-frequency atoms. These atoms are obtained from a single prototype function, by means of modulation, translation and warping. By warping we refer to the process of nonlinear evaluation according to a bijective, increasing function, the warping function. Besides showing that the resulting integral transforms fulfill certain basic, but essential properties, such as continuity and invertibility, we will show that a large subclass of warping functions gives rise to families of generalized coorbit spaces, i.e. Banach spaces of functions whose representations possess a certain localization. Furthermore, we obtain sufficient conditions for subsampled warped time-frequency systems to form atomic decompositions and Banach frames. To this end, we extend results previously presented by Fornasier and Rauhut to a larger class of function systems via a simple, but crucial modification. The proposed method allows for great flexibility, but by choosing particular warping functions we also recover classical time-frequency representations, e.g. $F(t) = ct$ provides the short-time Fourier transform and $F(t)=\log_a(t)$ provides wavelet transforms. This is illustrated by a number of examples provided in the manuscript