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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 GR ˜
UNBAUM 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 di#erential operators and found important applications of this phenomenon
in time­band 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 di#erential operators. They found important applica­
tions of this to time­band 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.

  

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

 

Collections: Mathematics