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Reflective prolate-spheroidal operators and the adelic Grassmannian by Casper, W. Riley; Grünbaum, F. Alberto; Yakimov, Milen; Zurrián, Ignacio is a scholarly article available to read on EtoBox.

What is Reflective prolate-spheroidal operators and the adelic Grassmannian about?

Beginning with the work of Landau, Pollak and Slepian in the 1960s on time-band limiting, commuting pairs of integral and differential operators have played a key role in signal processing, random matrix theory and integrable systems. Previously, such pairs were constructed by ad hoc methods, which worked because a commuting operator of low order could be found by a direct calculation. We describe a general approach to these problems that proves that every point $W$ of Wilson's infinite dimensional adelic Grassmannian $\mathrm Gr^ad$ gives rise to an integral operator $T_W$, acting on $L^2(\Gamma)$ for a contour $\Gamma\subset\mathbb C$, which reflects a differential operator $R(z,\partial_z)$ in the sense that $R(-z,-\partial_z)\circ T_W=T_W\circ R(w,\partial_w)$ on a dense subset of $L^2(\Gamma)$. By using analytic methods and methods from integrable systems, we show that the reflected differential operator can be constructed from the Fourier algebra of the associated bispectral function $\psi_W(x,z)$. The size of this algebra with respect to a bifiltration is in turn determined using algebro-geometric methods. Intrinsic properties of four involutions of the adelic Grassmannian n

Author

Casper, W. Riley; Grünbaum, F. Alberto; Yakimov, Milen; Zurrián, Ignacio

Published

2020

Language

EN