In the mid 80's it was conjectured that every bispectral meromorphic function ψ(x, y) gives rise to an integral operator K ψ (x, y) which possesses a commuting differential operator. This has been verified by a direct computation for several families of functions ψ(x, y) where the commuting differential operator is of order ≤ 6. We prove a general version of this conjecture for all self-adjoint bispectral functions of rank 1 and all self-adjoint bispectral Darboux transformations of the rank 2 Bessel and Airy functions. The method is based on a theorem giving an exact estimate of the second and first order terms of the growth of the Fourier algebra of each such bispectral function. From it we obtain a sharp upper bound on the order of the commuting differential operator for the integral kernel K ψ (x, y) and a fast algorithmic procedure for constructing the differential operator; unlike the previous examples its order is arbitrarily high. We prove that the above classes of bispectral functions are parametrized by infinite dimensional Grassmannians which are the Lagrangian loci of the Wilson adelic Grassmannian and its analogs in rank 2.
Beginning with the work of Landau, Pollak and Slepian in the 1960s on timeband 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 essentially 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 Gr ad gives rise to an integral operator TW , acting on L 2 (Γ) for a contour Γ ⊂ C, which reflects a differential operator with rational coefficients R(z, ∂z) in the sense that R(−z, −∂z) • TW = TW • R(w, ∂w) on a dense subset of L 2 (Γ). 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 ψW (x, z). The exact 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 above in place of plain commutativity. Furthermore, we prove that the time-band limited operators of the generalized Laplace transforms with kernels given by the rank one bispectral functions ψW (x, −z) always reflect a differential operator. A 90 • rotation argument is used to prove that the time-band limited operators of the generalized Fourier transforms with kernels ψW (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.that is, the eigenfunctions of the integral operatorThis is a numerically extremely unstable problem. Landau, Pollak and Slepian [43,32] bypassed this issue based on the property thatcommutes with EE * . The differential operator R(z, ∂ z ) is the "radial part" of the Laplacian in prolate-spheroidal coordinates, and (the joint) eigenfunctions of it and the integral operator EE * are known as the prolate-spheroidal functions, which can be computed numerically in very stable ways. The commuting pair can be traced back to Bateman [6,
In this paper, we demonstrate an elementary method for constructing new solutions to Bochner's problem for matrix differential operators from known solutions. We then describe a large family of solutions to Bochner's problem, obtained from classical solutions, which include several examples known from the literature. By virtue of the method of construction, we show how one may explicitly identify a generating function for the associated sequence of monic w-orthogonal matrix polynomials {p(x, n)}, as well as the associated algebra D(w) of all matrix differential operators for which the {p(x, n)} are eigenfunctions. We also include some general results on the structure of the algebra D(w).
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