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Strong Asymptotic Analysis of OLPs on the Unit Circle by Riemann-Hilbert Approach

  • Yufeng Wang
  • Yifeng Lu
  • Jinyuan DuEmail author
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Abstract

In this article, we will deal with the asymptotics of the monic orthogonal Laurent polynomials (OLPs) on the unit circle with respect to a strictly-positive analytic weight by Riemann-Hilbert approach. We first construct a matrix Riemann-Hilbert problem (RHP) which is the Fokas-Its-Kitaev characterization. Then, the strong asymptotic formulas of OLPs are obtained by employing Deift-Zhou steepest descent analysis. Furthermore, the asymptotic formulas of the leading coefficient and the trailing coefficient are simultaneously obtained.

Keywords

Orthogonal Laurent polynomial Riemann-Hilbert approach Riemann-Hilbert problem Strong asymptotics Cauchy-type integral operator 

Mathematics Subject Classification (2010)

42C05 41A35 30G30 
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Notes

Acknowledgements

While the corresponding author visited Free University Berlin in summer 2005 on basis of State Scholarship Fund Award of China, our group began to explore the application of Riemann-Hilbert approach, and made a debut [3]. During that time Professor H. Begehr carefully reviewed this manuscript and offered a lot of suggestions. All the authors are very grateful to Professor H. Begehr for his long-term support and help.

This work was supported by NNSF for Young Scholars of China (No. 11001206) and NNSF (No. 11171260).

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina
  2. 2.School of ScienceLinyi UniversityShandongChina

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