Local asymptotics for orthonormal polynomials on the unit circle via universality
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DORON S. LUBINSKY∗ Dedicated to L. Zalcman 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 ϕ (ζ (1+ z )) polynomials for μ. We show that for appropriate ζn ∈ J, { n ϕnn (ζn )n }n≥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.
1
Results
Let μ be a finite positive Borel measure on [−π, π) (or equivalently on the unit circle) with infinitely many points in its support. Then we may define orthonormal polynomials ϕn (z) = κn zn + · · · , κn > 0, n = 0, 1, 2, . . ., satisfying the orthonormality conditions π 1 ϕn (z)ϕm (z)dμ(θ) = δmn , 2π −π where z = eiθ . We shall often assume that μ is regular in the sense of Stahl, Totik and Ullmann [14], so that lim κn1/n = 1. n→∞
This is true if, for example, μ > 0 a.e. in [−π, π), but there are pure jump and pure singularly continuous measures that are regular. We denote the zeros of ϕn by {zjn }nj=1 . They lie inside the unit circle, and may not be distinct. ∗
Research supported by NSF grant DMS1800251.
285 ´ JOURNAL D’ANALYSE MATHEMATIQUE, Vol. 141 (2020) DOI 10.1007/s11854-020-0121-8
286
D. S. LUBINSKY
The n-th reproducing kernel for μ is Kn (z, u) =
n−1
ϕj (z)ϕj (u).
j=0
One of the key limits in random matrix theory, the so-called universality limit [4], [5], [7], [8], [13], [16], [17], can be cast in the following form for measures on the unit circle [6, Thm. 6.3, p. 559]: Theorem A. Let μ be a finite positive Borel measure on the unit circle that is regular. Let J ⊂ (−π, π) be compact, and such that μ is absolutely continuous in an open set containing J. Assume moreover, that μ is positive and continuous at each point of J. Then uniformly for θ ∈ J, z = eiθ and a, b in compact subsets of the complex plane, we have (1.1) where S(t) =
lim
n→∞
Kn (z(1 +
i2πa i2πb¯ n ), z(1 + n ))
Kn (z, z)
= eiπ(a−b) S(a − b),
sin πt πt .
There are several refinements and generalizations of this result; see, for example, [11], [13], [16], [17]. In this paper, we shall use the universality limit to establish “local” asymptotics for the ratio ϕn (z(1 + un ))/ϕn (z) with u as our variable. Analogous results for orthogonal polynomials associated with measures on compact subsets of the real line were established in [9], [10]. In [9], we showed that if μ is a regular measure on [−1, 1] for which μ (x)(1 − x)−α has a finite positive limit as x → 1−, then the orthonormal polynomials {pn } for μ satisfy, uniformly for z in compact subsets of C, z2 pn (1 − 2n J ∗ (z) 2) = ∗α , lim n→∞ pn (1) Jα (0) where Jα∗ (z) = Jα (z)/zα is the normalized Bessel function of order α. In [10], we showed that if μ is a regular measure with compact support in the real line, and in some closed subinterval J of the support, μ is absolutely continuou
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