Harmonic Analysis and Applications

June 4-8, 2018


"Poisson Summation, Selberg Trace, and Sampling on General Manifolds"

Casey, Stephen

Sampling 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.

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