Math @ Duke

Publications [#244285] of Xin Zhou
Papers Published
 with McLaughlin, KTR; Vartanian, AH; Zhou, X, Asymptotics of laurent polynomials of odd degree orthogonal with respect to varying exponential weights,
Constructive Approximation, vol. 27 no. 2
(2008),
pp. 149202, ISSN 01764276 [doi]
(last updated on 2018/09/19)
Abstract: Let Λ ℝ denote the linear space over ℝ spanned by zk k ℤ. Define the (real) inner product Ċ,Ċ L : Λ ℝ× Λ ℝ ℝ, (f,g) ∫ℝ f(s)g(s) exp( N V(s)) ds, N ℕ, where V satisfies: (i) V is real analytic on ℝ 0; (ii) lim x (V(x)/ln(x2} + 1)) = + and (iii) limx 0(V(x)/ln (x2} + 1)) = +. Orthogonalisation of the (ordered) base 1,z1,z,z2z2},z k},zk with respect to , {{ L}} yields the even degree and odd degree orthonormal Laurent polynomials φ{m}(z)m=0: φ2n(z) = ξ(2n)zn + + ξ(2n)nzn ξ(2n)n > 0, and φ{2n+1}(z) = ξ(2n+1)n1zn1 + + ξ(2n+1)nz n ξ(2n+1)n1 > 0. Define the even degree and odd degree monic orthogonal Laurent polynomials: π2n(z) := (ξ(2n)n1} φ2n(z) and π{2n+1}(z) := (ξ(2n+1)n11 φ2n+1(z). Asymptotics in the doublescaling limit N,n such that N,n = 1 + o(1) of π2n+1(z) (in the entire complex plane), ξ(2n+1)n1, and φ2n+1(z)(in the entire complex plane) are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a matrix RiemannHilbert problem on ℝ, and then extracting the largen behaviour by applying the nonlinear steepestdescent method introduced in [1] and further developed in [2],[3]. © 2007 Springer.


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