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Centre de Recherches Mathematiques CRM Proceedings and Lecture Notes

Summary: Centre de Recherches Math´ematiques
CRM Proceedings and Lecture Notes
Volume 37, 2004
The Prolate Spheroidal Phenomenon
as a Consequence of Bispectrality
F. Alberto Gr¨unbaum and Milen Yakimov
Abstract. In this paper we announce that a very large class of integral op-
erators derived from bispectral algebras of rank 1 and 2 (parametrized by
Lagrangian Grassmannians of infinitely large size) posses commuting differen-
tial operators. The examples of Landau, Pollak, Slepian, and Tracy, Widom,
used in time-band limiting and random matrix theory arise as special cases of
this result.
1. Introduction
It was discovered by Landau, Pollak, Slepian, and Tracy, Widom, that cer-
tain integral operators associated to the Airy and Bessel special functions possess
commuting differential operators. They found important applications of this to
time-band limiting, and to the study of asymptotics of Fredholm determinants,
relevant to scaling limits of random matrix models. We call this phenomenon the
prolate spheroidal phenomenon.
On the other hand, the problem of bispectrality was posed [7] about 20 years


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics