Time-frequency representations for nonlinear frequency scales — Coorbit spaces and discretization

The fixed time-frequency resolution of the short-time Fourier transform has often been considered a major drawback. In this contribution we review recent results on a class of time-frequency transforms that adapt to a large class of frequency scales in the same sense that wavelet transforms are adapted to a logarithmic scale. In particular, we show that each transform in this class of warped time-frequency representations is a tight continuous frame satisfying orthogonality relations similar to Moyal's formula. Moreover, they satisfy the prerequisites of generalized coorbit theory, giving rise to coorbit spaces and associated discrete representations, i.e. atomic decompositions and Banach frames.