Poisson Summation and Selberg Trace

Sampling theory is a fundamental area of study in harmonic analysis and signal and image processing. We look at the key role the Poisson Summation Formula plays in sampling, and show its connection to the Selberg Trace Formula. We discuss harmonic analysis in both Euclidean and non-Euclidean spaces, and then focus on the connection of the Poisson Summation Formula to the Selberg Trace Formula in non-Euclidean settings.