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.