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Publications [#244285] of Xin Zhou

Papers Published

1. 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 (March, 2008), pp. 149-202, Springer Nature, ISSN 0176-4276 [doi]
(last updated on 2021/05/15)

Abstract:
Let Λ denote the linear space over ℝ spanned by z k ℤ. Define the (real) inner product Ċ,Ċ L : Λ × Λ ℝ, (f,g) ∫ 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(x } + 1)) = + and (iii) limx 0(V(x)/ln (x } + 1)) = +. Orthogonalisation of the (ordered) base 1,z ,z,z z },z },z with respect to , {{ L}} yields the even degree and odd degree orthonormal Laurent polynomials φ{m}(z) : φ (z) = ξ z + + ξ z ξ > 0, and φ (z) = ξ z + + ξ z ξ > 0. Define the even degree and odd degree monic orthogonal Laurent polynomials: π (z) := (ξ } φ (z) and π (z) := (ξ φ (z). Asymptotics in the double-scaling limit N,n such that N,n = 1 + o(1) of π (z) (in the entire complex plane), ξ , and φ (z)(in the entire complex plane) are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on ℝ, and then extracting the large-n behaviour by applying the non-linear steepest-descent method introduced in [1] and further developed in [2],[3]. © 2007 Springer. ℝ k ℝ ℝ 2 2 -1 -2 2 -k k (2n) -n (2n) n (2n) {2n+1} (2n+1) -n-1 (2n+1)n n (2n+1) (2n) -1 {2n+1} (2n+1) -1 (2n+1) ℝ m=0 2n n n -n-1 -n-1 2n n 2n -n-1 2n+1 2n+1 -n-1 2n+1

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