**"Poisson Summation, Selberg Trace, and Sampling on General Manifolds"**
#### Casey, StephenSampling theory is a fundamental area of study in harmonic analysis
and signal and image processing. Our talk will connect sampling theory
with the geometry of the signal and its domain. It is relatively easy to
demonstrate this connection in Euclidean spaces, but one quickly gets into
open problems when the underlying space is not Euclidean. In particular,
we discuss spherical geometry, hyperbolic geometry, and the geometry of
general surfaces.
We first look at the key role the Poisson Summation Formula plays in
sampling, and show its connection to the Selberg Trace Formula. We then
focus on the connection of the Poisson Summation Formula to the Selberg
Trace Formula in non-Euclidean settings.
There are numerous motivations for extending sampling to non-Euclidean
geometries. Applications of sampling in spherical and hyperbolic geometries
are showing up areas from EIT to cosmology. Sampling in spherical geometry
has been analyzed by many authors, e.g., Driscoll, Healy, Keiner, Kunis,
McEwen, Potts, and Wiaux, and brings up questions about tiling the
sphere. Irregular sampling of band-limited functions by iteration in
hyperbolic space is possible, as shown by Feichtinger and Pesenson.
In Euclidean space, the minimal sampling rate for Paley-Wiener
functions on R^d, the Nyquist rate, is a function of the band-width.
No such rate has yet been determined for hyperbolic or spherical spaces.
We look to develop a structure for the tiling of frequency spaces in
both Euclidean and non-Euclidean domains. In particular, we develop
an approach to determine Nyquist tiles and sampling groups
for spherical and hyperbolic space. We then connect this to arbitrary orientable
analytic surfaces using Uniformization. |