Asymptotics of laurent polynomials of odd degree orthogonal with respect to varying exponential weights

Kenneth D T Mclaughlin, A. H. Vartanian, X. Zhou

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

Let Λ denote the linear space over ℝ spanned by zk 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(x2} + 1)) = + and (iii) limx 0(V(x)/ln (x2} + 1)) = +. Orthogonalisation of the (ordered) base 1,z-1,z,z-2z2},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)z-n + + ξ(2n)nzn ξ(2n)n > 0, and φ{2n+1}(z) = ξ(2n+1) -n-1z-n-1 + + ξ(2n+1)nz n ξ(2n+1)-n-1 > 0. Define the even degree and odd degree monic orthogonal Laurent polynomials: π2n(z) := (ξ(2n)n-1} φ2n(z) and π{2n+1}(z) := (ξ(2n+1)-n-1-1 φ2n+1(z). Asymptotics in the double-scaling limit N,n such that N,n = 1 + o(1) of π2n+1(z) (in the entire complex plane), ξ(2n+1)-n-1, and φ2n+1(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].

Original languageEnglish (US)
Pages (from-to)149-202
Number of pages54
JournalConstructive Approximation
Volume27
Issue number2
DOIs
StatePublished - Mar 2008

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Exponential Weights
Laurent Polynomials
Orthogonal Laurent Polynomials
Monic
Odd
Polynomials
Argand diagram
Entire
Steepest descent method
Orthonormal Polynomials
Steepest Descent Method
Orthogonalization
Riemann-Hilbert Problem
Scaling Limit
Linear Space
Scalar, inner or dot product
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Keywords

  • Asymptotics
  • Equilibrium measures
  • Hankel determinants
  • Laurent polynomials
  • Laurent-Jacobi matrices
  • Riemann-Hilbert problems

ASJC Scopus subject areas

  • Mathematics(all)
  • Analysis

Cite this

Asymptotics of laurent polynomials of odd degree orthogonal with respect to varying exponential weights. / Mclaughlin, Kenneth D T; Vartanian, A. H.; Zhou, X.

In: Constructive Approximation, Vol. 27, No. 2, 03.2008, p. 149-202.

Research output: Contribution to journalArticle

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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,z-1,z,z-2z2},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)z-n + + ξ(2n)nzn ξ(2n)n > 0, and φ{2n+1}(z) = ξ(2n+1) -n-1z-n-1 + + ξ(2n+1)nz n ξ(2n+1)-n-1 > 0. Define the even degree and odd degree monic orthogonal Laurent polynomials: π2n(z) := (ξ(2n)n-1} φ2n(z) and π{2n+1}(z) := (ξ(2n+1)-n-1-1 φ2n+1(z). Asymptotics in the double-scaling limit N,n such that N,n = 1 + o(1) of π2n+1(z) (in the entire complex plane), ξ(2n+1)-n-1, and φ2n+1(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].",
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N2 - 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,z-1,z,z-2z2},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)z-n + + ξ(2n)nzn ξ(2n)n > 0, and φ{2n+1}(z) = ξ(2n+1) -n-1z-n-1 + + ξ(2n+1)nz n ξ(2n+1)-n-1 > 0. Define the even degree and odd degree monic orthogonal Laurent polynomials: π2n(z) := (ξ(2n)n-1} φ2n(z) and π{2n+1}(z) := (ξ(2n+1)-n-1-1 φ2n+1(z). Asymptotics in the double-scaling limit N,n such that N,n = 1 + o(1) of π2n+1(z) (in the entire complex plane), ξ(2n+1)-n-1, and φ2n+1(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].

AB - 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,z-1,z,z-2z2},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)z-n + + ξ(2n)nzn ξ(2n)n > 0, and φ{2n+1}(z) = ξ(2n+1) -n-1z-n-1 + + ξ(2n+1)nz n ξ(2n+1)-n-1 > 0. Define the even degree and odd degree monic orthogonal Laurent polynomials: π2n(z) := (ξ(2n)n-1} φ2n(z) and π{2n+1}(z) := (ξ(2n+1)-n-1-1 φ2n+1(z). Asymptotics in the double-scaling limit N,n such that N,n = 1 + o(1) of π2n+1(z) (in the entire complex plane), ξ(2n+1)-n-1, and φ2n+1(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].

KW - Asymptotics

KW - Equilibrium measures

KW - Hankel determinants

KW - Laurent polynomials

KW - Laurent-Jacobi matrices

KW - Riemann-Hilbert problems

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