Karen Shuman writes:
As for the signal processing, I would like to see the different types of
basis elements which my weird group produces. These basis elements are
functions of 2 complex variables and they map into the complex numbers, so
it is not completely trivial to see that the group is actually doing to
different types of functions. Even simpler but also as interesting
(perhaps the first thing to do?) is the fundamental domain for the
discrete group action. It would be nice to be able to see it--it lives
in the product of a half complex plane and a complex plane. One time I
drew by hand several different snapshots of where the boundary of the
domain goes under different group elements (non-discrete) and it was
tedious, but it also looked pretty cool. Part of the action is going on
in hyperbolic space but the other half is going on in Euclidean space, so
things get twisted around (at least my little drawings indicated this.)
None of this sounds like signal processing, but as I was thinking about
my problem the thing that bugged me all the way through my own thesis
was that these things were hard to visualize, and this would be a good
opportunity to see some of them. Not ground-breaking research, since I
imagine experts in the Jacobi group know these things intuitively, but
I don't and when I talked about what I was doing no one else could see
it very clearly either.
Then there is the actual signal processing question---if this group will
work, what types of functions, which group elements should be used to
generate the functions, etc---I think all of these questions could be
answered much better if we could see things first.