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Global Asymptotics of Orthogonal Polynomials Associated with a Generalized Freud Weight

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Chinese Annals of Mathematics,Series B 2018年第3期39卷 553-596页

作者： Zhi-Tao WEN roderick wong Shuai-Xia XU Department of Mathematics Taiyuan University of TechnologyTaiyuan 030024China Liu Bie Ju Centre for Mathematical Sciences City University of Hong KongHong KongChina Institut Franco-Chinois de l’Energie Nucleaire Sun Yat-sen UniversityGuangzhou 510275China

In this paper,the authors consider the asymptotic behavior of the monic polynomials orthogonal with respect to the weight function w(x) = |x|^(2α)e^(-(x^4+tx^2)),x ∈ R,where α is a constant larger than -1/2 and t is any real number. They consider this problem in three separate cases:(i) c >-2,(ii) c =-2,and(iii) c <-2,where c := t N^(-1/2) is a constant,N = n + α and n is the degree of the polynomial. In the first two cases,the support of the associated equilibrium measure μ_t is a single interval,whereas in the third case the support of μ_t consists of two intervals. In each case,globally uniform asymptotic expansions are obtained in several regions. These regions together cover the whole complex plane. The approach is based on a modified version of the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou(1993).

关键词：多项式直角重量盒子常数实数

Error Bounds for Uniform Asymptotic Expansions-Modified Bessel Function of Purely Imaginary Order

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Chinese Annals of Mathematics,Series B 2010年第5期31卷 759-780页

作者： Wei SHI roderick wong Wuhan Institute of Physics and Mathematics The Chinese Academy of Sciences Wuhan 430071 China. Department of Mathematics City University of Hong Kong Tat Chee Avenue Kowloon Hong Kong China.

The authors modify a method of Olde Daalhuis and Temme for representing the remainder and coefficients in Airy-type expansions of *** using a class of rational functions,they express these quantities in terms of Cauchy-type integrals;these expressions are natural generalizations of integral representations of the coe?cients and the remainders in the Taylor expansions of analytic *** using the new representation,a computable error bound for the remainder in the uniform asymptotic expansion of the modified Bessel function of purely imaginary order is derived.

关键词：渐近展开误差界虚函数塞尔高阶积分计算泰勒展开式有理函数

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浅谈渐近分析献给徐利治教授95华诞

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中国科学：数学 2015年第9期45卷 1363-1382页

作者：王世全香港城市大学中国香港

本文旨在对20世纪后半叶关于渐近分析理论的最新进展作全方位的概述,其中包括一些经典领域,如积分的渐近逼近、微分方程的渐近解和奇异摄动理论;也涵盖了一些新兴领域,如差分方程的渐近理论和Riemann-Hilbert方法.

关键词：最速下降法 WKB(Wentzel-Kramers-Brillouin)方法奇异摄动理论差分方程 Riemann-Hilbert方法