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 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 doublescaling 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 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. ℝ k ℝ ℝ 2 2 1 2 2 k k (2n) n (2n) n (2n) {2n+1} (2n+1) n1 (2n+1)n n (2n+1) (2n) 1 {2n+1} (2n+1) 1 (2n+1) ℝ m=0 2n n n n1 n1 2n n 2n n1 2n+1 2n+1 n1 2n+1


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