Continuous Gabor transform for a class of non-abelian groups (Q1932159)

The authors generalize the notion of continuous Gabor transform from abelian groups to a class of nonabelian groups, which are second-countable, unimodular and of type I. Two well-known facts from Gabor theory, Plancherel and inversion theorems, are extended to this setting. Finally, the results are checked for the Heisenberg group and the matrix group \(\operatorname{SL}(2,\mathbb{R})\), as outstanding examples of nonabelian second-countable unimodular type I groups.