F. Alberto Grünbaum: Time and band limiting for exceptional orthogonal polynomials

Atelier sur Le rôle des systèmes intégrables - Atelier dédié à John Harnad /Workshop on the role of integrable systems - Workshop dedicated to John Harnad: novembre/November 14-25 & décembre/December 05-09

Décembre/December 08 :
Webpage: https://www.crmath.ca/en/activities/#...
Schedule: http://www.crm.umontreal.ca/2022/Harn...

The phenomenon of time-and-band limiting goes back to work by Slepian, Landau and Pollak at Bell labs in the 1960s. One has a naturally appearing integral operator that by some miracle commutes with an explicit differential operator, the prolate spheroidal wave operator. A natural question is what happens when Fourier analysis is replaced by expansions on basis different from the exponentials. This gave rise to the "bispectral problem" posed and solved in a paper with H. Duistermaat. There are surprising connections with the KdV equation and its master symmetries and many other topics such as very simple monodromy. There are important contributions by G. Wilson, B. Bakalov, E. Horozov, M.Yakimov, A. Kasman, M. Rothstein to the bispectral problem. The first attack on the matrix valued version of the bispectral problem is in the Berkeley thesis of J. Zubelli who found a role for the AKNS systems. The discrete-continuous case is treated in a few papers with L. Haine.The are extensions to the matrix valued case in papers with I. Pacharoni, J. Tirao, P. Iliev and A. Duran. I have been trying to return to the original time-and-band problem in the scalar and matrix valued setup. There are papers with I. Pacharoni, I. Zurrian and M. Castro, L. Vinet and A Zhedanov. There are connections with Random matrix theory as in the work of M. Mehta, C. Tracy and H. Widom. We have an ongoing project and a few papers already out with W. Casper, M. Yakimov and I. Zurrian extending the original work in several directions. Very recently, in joint work with M. Castro we have looked at the case of exceptional orthogonal polynomials a nice instance of a bispectral situation. I am happy to notice that we have found good use of some tools developed by my former students M. Perlstadt, R. Perline and M. Reach. Time and band limiting plays an intriguing role in work that A. Connes has been doing for quite some time in connection with Riemann's zeta function, as well as recent work in connection with the Bethe Ansatz by L. Vinet and some of his friends, such as P. Baseilhac, R. Pimenta, P. Bernard and N. Campre. There are nice papers in a CRM volume at CRM edited by J. Harnad and A. Kasman.