I have been looking for an analog of the Fourier transform for groupoids coming from tilings (which are generally principal and r-discrete), however all the generalizations I have found assume that the groupoid:
is compact (http://arxiv.org/pdf/math/0308260.pdf).
is a group bundle (http://www.ams.org/journals/tran/1996-348-09/S0002-9947-96-01610-8/S0002-9947-96-01610-8.pdf) pg 3632
is decomposable (http://cdn.intechopen.com/pdfs/15159/InTech-Fourier_transform_on_group_like_structures_and_applications.pdf) section 7.
Are there any versions of the Fourier transform for a principal r-discrete groupoid?