  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 naturally lead us to
consider the reflecting property in place of plain commutativity. Furthermore,
we prove that the time-band limited operators of the generalized Laplace
transforms with kernels given by all rank one bispectral functions
$\psi_W(x,-z)$ reflect a differential operator. A $90^\circ$ rotation argument
is used to prove that the time-band limited operators of the generalized
Fourier transforms with kernels $\psi_W(x,iz)$ admit a commuting differential
operator. These methods produce vast collections of integral operators with
prolate-spheroidal properties, associated to the wave functions of all rational
solutions of the KP hierarchy vanishing at infinity, introduced by Krichever in
the late 1970s.
