Applications of a new formula for OPUC with periodic Verblunsky coefficients

Simanek, Brian

doi:10.1007/s40687-018-0165-x

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Hypothesis 17 Consider Matrix-valued Functions1

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2021

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“…, n − 1}, which would be related to the CMV basis, this factor would not be needed. Of course, this prefactor can be simplified in cases when k n (e iξ , e iξ )/n has a limit, as is the case in prior results [15,24,32,41]. The linear behavior of the Christoffel function is studied in great generality in [49].…”

Section: Hypothesis 17 Consider Matrix-valued Functionsmentioning

confidence: 94%

An approach to universality using Weyl m-functions

Eichinger,

Lukić,

Simanek

2021

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We describe an approach to universality limits for orthogonal polynomials on the real line which is completely local and uses only the boundary behavior of the Weyl m-function at the point. We show that bulk universality of the Christoffel-Darboux kernel holds for any point where the imaginary part of the m-function has a positive finite nontangential limit. This approach is based on studying a matrix version of the Christoffel-Darboux kernel and the realization that bulk universality for this kernel at a point is equivalent to the fact that the corresponding m-function has normal limits at the same point. Our approach automatically applies to other self-adjoint systems with 2 × 2 transfer matrices such as continuum Schrödinger and Dirac operators. We also obtain analogous results for orthogonal polynomials on the unit circle.

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