  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.

Functional Analysis