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We present a novel family of continuous linear time-frequency transformsadapted to a multitude of (nonlinear) frequency scales. Similar to classicaltime-frequency or time-scale representations, the representation coefficientsare obtained as inner products with the elements of a continuously indexedfamily of time-frequency atoms. These atoms are obtained from a singleprototype function, by means of modulation, translation and warping. By warpingwe refer to the process of nonlinear evaluation according to a bijective,increasing function, the warping function. Besides showing that the resultingintegral transforms fulfill certain basic, but essential properties, such ascontinuity and invertibility, we will show that a large subclass of warpingfunctions gives rise to families of generalized coorbit spaces, i.e. Banachspaces of functions whose representations possess a certain localization.Furthermore, we obtain sufficient conditions for subsampled warpedtime-frequency systems to form atomic decompositions and Banach frames. To thisend, we extend results previously presented by Fornasier and Rauhut to a largerclass of function systems via a simple, but crucial modification. The proposedmethod allows for great flexibility, but by choosing particular warpingfunctions we also recover classical time-frequency representations, e.g. $F(t)= ct$ provides the short-time Fourier transform and $F(t)=\log_a(t)$ provideswavelet transforms. This is illustrated by a number of examples provided in themanuscript.

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