The prolate spheroidal phenomena and bispectrality

Milen Yakimov (milen@math.cornell.edu)
Cornell University
Department of Mathematics
Ithaca, NY 14850 USA

Abstract
The problem of bispectrality was posed 20 years ago by Alberto Grunbaum as a tool to understand the prolate spheroidal phenomena of Landau, Pollak, and Slepian (existence of a commuting differential operator for some integral operators). Since the 60's the latter found numerous applications, e.g. in random matrix theory it is used to study asymptotics of Fredholm determinants. At the same time, to the best of our knowledge, the two problems remained isolated except for few common examples, and the prolated spheroidal phenomena is known for a rather small class of integral operators. In this talk we will try to explain a very general connection between the two problems: All self-adjoint bispectral algebras of rank 1 and 2 (which is a very large family) lead to integral operators which posses the prolate spheroidal property. (The latter are in the class of "integrable integral operators"). This is a joint work with F. A. Grunbaum (Berkeley).