THE PROLATE SPHEROIDAL PHENOMENON AND BISPECTRAL ALGEBRAS OF ORDINARY DIFFERENTIAL OPERATORS

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