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
(March, 2008),
pp. 149202, Springer Nature, ISSN 01764276 [doi]
(last updated on 2019/05/22)
Abstract: Let Λ ℝ denote the linear space over ℝ spanned by z k 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(x 2 } + 1)) = + and (iii) limx 0(V(x)/ln (x 2 } + 1)) = +. Orthogonalisation of the (ordered) base 1,z 1 ,z,z 2 z 2 },z k },z k with respect to , {{ L}} yields the even degree and odd degree orthonormal Laurent polynomials φ{m}(z) m=0 : φ 2n (z) = ξ (2n) z n + + ξ (2n)n z n ξ (2n)n > 0, and φ {2n+1} (z) = ξ (2n+1)n1 z n1 + + ξ (2n+1)n z n ξ (2n+1)n1 > 0. Define the even degree and odd degree monic orthogonal Laurent polynomials: π 2n (z) := (ξ (2n)n1 } φ 2n (z) and π {2n+1} (z) := (ξ (2n+1)n11 φ 2n+1 (z). Asymptotics in the doublescaling limit N,n such that N,n = 1 + o(1) of π 2n+1 (z) (in the entire complex plane), ξ (2n+1)n1 , and φ 2n+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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