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Local asymptotics for orthonormal polynomials on the unit circle via universality

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Abstract

Let µ be a positive measure on the unit circle that is regular in the sense of Stahl, Totik, and Ullmann. Assume that in some subarc J, µ is absolutely continuous, while µ′ is positive and continuous. Let {φn} be the orthonormal polynomials for µ. We show that for appropriate ζnJ, \({{\rm{\{ }}{{{\varphi _n}({\zeta _n}(1 + {\textstyle{z \over n}}))} \over {{\varphi _n}({\zeta _n})}}{\rm{\}}}_{n \ge 1}}\) is a normal family in compact subsets of ℂ. Using universality limits, we show that limits of subsequences have the form ez + C(ez − 1) for some constant C. Under additional conditions, we can set C = 0.

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Correspondence to Doron S. Lubinsky.

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Dedicated to L. Zalcman

Research supported by NSF grant DMS1800251.

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Lubinsky, D.S. Local asymptotics for orthonormal polynomials on the unit circle via universality. JAMA 141, 285–304 (2020). https://doi.org/10.1007/s11854-020-0121-8

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  • DOI: https://doi.org/10.1007/s11854-020-0121-8

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