Pointwise Asymptotics for Orthonormal Polynomials at the Endpoints of the Interval via Universality
Pointwise Asymptotics for Orthonormal Polynomials at the Endpoints of the Interval via Universality
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DOI：
10.1093/imrn/rny042
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发表时间：
2020-02
影响因子：
1
通讯作者：
D. Lubinsky
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文献类型：
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作者：
D. Lubinsky
我们证明正交多项式的普遍性极限和界限意味着正交多项式在正交性区间的端点处的点向渐近。因此，我们证明，如果 μ 是 [−1, 1] 上支持的正则测度，并且在 1 的邻域中，μ 是绝对连续的，而对于某些 α > −1，μ′ (t) = h (t) (1− t)，其中 h (t) → 1 为 t → 1−，则相应的正交多项式 {pn} 满足渐近 lim n→∞ pn ( 1− z 2 2n2 ) pn (1) = J* α (z) J* α (0) 在平面的紧子集中一致。这里 J* α (z) = Jα (z) /z α 是 α 阶的归一化贝塞尔函数。这些是迄今为止此类终点渐近的最一般条件。 1. 结果 令 μ 为具有紧支撑的有限正 Borel 测度，包含无限多个点。然后我们可以定义正交多项式 pn (x) = γnx n + ..., γn > 0, n = 0, 1, 2, ... 满足正交性条件 ∫ pnpmdμ = δmn。我们用 xnn < xn−1,n < ... < x2n < x1n 表示 pn 的零点。 {pn} 满足三项递推关系 xpn−1 (x) = anpn (x) + bn−pn− (x) + an−1pn−2 (x) ，其中 an = γn−1 γn 且 bn ∈ R。当 n → Infini 时 pn 的渐近是一个被广泛研究的课题，并且有许多应用。 μ 支撑内部的渐近与边缘或外部的渐近有很大不同。在本文中，我们关注边缘的渐近。最著名的此类渐近公式是经典雅可比多项式的 Mehler-Heine 公式 { P (α,β) n } ，该公式与雅可比权重 (1.1) w (x) = (1− x) (1 + x) , x ∈ (−1, 1) 正交，日期：2018 年 1 月 2 日。 1
We show that universality limits and bounds for orthonormal polynomials 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), where h (t) → 1 as t → 1−, then the corresponding orthonormal polynomials {pn} satisfy the asymptotic lim n→∞ pn ( 1− z 2 2n2 ) pn (1) = J∗ α (z) J∗ α (0) uniformly in compact subsets of the plane. Here J∗ α (z) = Jα (z) /z α is the normalized Bessel function of order α. These are by far the most general conditions for such endpoint asymptotics. 1. Results Let μ be a finite positive Borel measure with compact support, contianing infinitely many points. Then we may define orthonormal polynomials pn (x) = γnx n + ..., γn > 0, n = 0, 1, 2, ... satisfying the orthonormality conditions ∫ pnpmdμ = δmn. We denote the zeros of pn by xnn < xn−1,n < ... < x2n < x1n. The {pn} satisfy the three term recurrence relation xpn−1 (x) = anpn (x) + bn−pn− (x) + an−1pn−2 (x) , where an = γn−1 γn and bn ∈ R. Asymptotics for pn as n → ∞ are a much studied subject, and have numerous applications. The asymptotic in the interior of the support of μ, is quite different from that at the edges, or in the exterior. In this paper, we focus on asymptotics at the edges. The best known such asymptotic is the Mehler-Heine formula for classical Jacobi polynomials { P (α,β) n } , which are orthogonal with respect to the Jacobi weight (1.1) w (x) = (1− x) (1 + x) , x ∈ (−1, 1) , Date : January 2, 2018. 1