Integral operators, bispectrality and growth of Fourier algebras

W. Riley Casper; Milen T. Yakimov

doi:10.1515/crelle-2019-0031

Article

Integral operators, bispectrality and growth of Fourier algebras

Published/Copyright: October 1, 2019

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2020 Issue 766

Abstract

In the mid 1980s 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 ) leading to 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.

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1601862

Award Identifier / Grant number: DMS-1901830

Funding statement: The research of W. Riley Casper was supported by an AMS-Simons Travel Grant and that of Milen T. Yakimov by NSF grants DMS-1601862, DMS-1901830 and Bulgarian Science Fund grant DN02/05.

Acknowledgements

We are grateful to F. Alberto Grünbaum for his insight, suggestions and support throughout the different stages of this project. We are also grateful to the anonymous referee for the very helpful comments, observations, suggestions, and corrections, which greatly improved the quality of the manuscript.

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Received: 2018-10-25

Revised: 2019-07-19

Published Online: 2019-10-01

Published in Print: 2020-09-01

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