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Title: Pointwise Asymptotics for Orthonormal Polynomials at the Endpoints of the Interval Via Universality
We show that universality limits and other bounds imply pointwise asymptotics for orthonormal polynomials at the endpoints of the interval of orthonormality. As a consequence, we show that if μ is a regular measure supported on [−1, 1], and in a neighborhood of 1, μ is absolutely continuous, while for some α > −1, μ (t) = h (t)(1 − t) α, where h (t) → 1 as t → 1−, then the corresponding orthonormal polynomials {pn} satisfy the asymptotic limn→∞pn1 − z22n2pn (1) = J∗α (z)J∗α (0) uniformly in compact subsets of the plane. Here J∗α (z) = Jα (z) /zα is the normalized Bessel function of order α. These are by far the most general conditions for such endpoint asymptotics  more » « less
Award ID(s):
1800251
PAR ID:
10092098
Author(s) / Creator(s):
Date Published:
Journal Name:
International mathematics research notices
Volume:
2019
ISSN:
1073-7928
Page Range / eLocation ID:
1-22
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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