Eichinger, B. (2023). Necessary and Sufficient conditions for Universality limits. In Book of abstracts : International Conference on Spectral Theory and Approximation (pp. 3–3).
Book of abstracts : International Conference on Spectral Theory and Approximation
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Date (published):
8-Aug-2023
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Event name:
International Conference on Spectral Theory and Approximation 2023
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Event date:
14-Aug-2023 - 18-Aug-2023
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Event place:
Lund, Sweden
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Number of Pages:
1
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Keywords:
universality limits; orthogonal polynomials; canonical systems
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Abstract:
In this talk I present necessary and sufficient conditions for universality limits for orthogonal polynomials on the real line. One of our results is that the Christoffel-Darboux kernel has sine kernel asymptotics at a point ξ, with regularly varying scaling, if and only if the orthogonality measure (spectral measure) has a unique tangent measure at ξ and that is the Lebesgue measure. This includes all prior results with absolutely continuous or singular measures. In fact, sine kernel asymptotics is a special case of a more general theory which also includes hard edge universality limits; we show that the Christoffel-Darboux kernel has a regularly varying scaling limit if and only if the orthogonality measure has a unique tangent measure at ξ and that is not the point mass at ξ. In this case the limit kernel is expressible in terms of confluent hypergeometric functions. This talk is based on a joint work in progress with Milivoje Lukic and Harald Woracek.
