The $\alpha$-Modulation Transform: Admissibility, Coorbit Theory and   Frames of Compactly Supported Functions
  The $\alpha$-modulation transform is a time-frequency transform generated by square-integrable representations of the affine Weyl-Heisenberg group modulo suitable subgroups. In this paper we prove new conditions that guarantee the admissibility of a given window function. We also show that the generalized coorbit theory can be applied to this setting, assuming specific regularity of the windows. This then yields canonical constructions of Banach frames and atomic decompositions in $\alpha$-modulation spaces. In particular, we prove the existence of compactly supported (in time domain) vectors that are admissible and satisfy all conditions within the coorbit machinery, which considerably go beyond known results. 
Michael Speckbacher, Dominik Bayer, Stephan Dahlke and Peter Balazs
math.FA