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THE PROLATE SPHEROIDAL PHENOMENON AND BISPECTRAL ALGEBRAS OF ORDINARY DIFFERENTIAL
 

Summary: THE PROLATE SPHEROIDAL PHENOMENON AND
BISPECTRAL ALGEBRAS OF ORDINARY DIFFERENTIAL
OPERATORS
F. ALBERTO GRšUNBAUM AND MILEN YAKIMOV
Abstract. Landau, Pollak, Slepian, and Tracy, Widom discovered that cer-
tain integral operators with so called Bessel and Airy kernels possess commut-
ing differential operators and found important applications of this phenomenon
in time-band limiting and random matrix theory. In this paper we announce
that very large classes of integral operators derived from bispectral algebras
of rank 1 and 2 (parametrized by lagrangian grassmannians of infinitely large
size) have this property. The above examples come from special points in these
grassmannians.
1. Introduction
It was discovered by Landau, Pollak, Slepian [23, 18, 19, 22] and Tracy, Widom
[25, 26] that certain integral operators associated to the Airy and Bessel special
functions posses commuting differential operators. They found important applica-
tions 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 [8] about 20 years

  

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

 

Collections: Mathematics