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arXiv · 1607.01962

The CMV bispectral problem

F. A. Grünbaum·L. Velázquez
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Abstract

A classical result due to Bochner classifies the orthogonal polynomials on the real line which are common eigenfunctions of a second order linear differential operator. We settle a natural version of the Bochner problem on the unit circle which answers a similar question concerning orthogonal Laurent polynomials and can be formulated as a bispectral problem involving CMV matrices. We solve this CMV bispectral problem in great generality proving that, except the Lebesgue measure, no other one on the unit circle yields a sequence of orthogonal Laurent polynomials which are eigenfunctions of a linear differential operator of arbitrary order. Actually, we prove that this is the case even if such an eigenfunction condition is imposed up to finitely many orthogonal Laurent polynomials.

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F. A. Grünbaum, L. Velázquez. 2016-09-05. The CMV bispectral problem. https://doi.org/10.1093/imrn%2Frnw186

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