Asymptotic Behavior and Zero Distribution of Polynomials Orthogonal with Respect to Bessel Functions

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• 作者：Alfredo Deaño ; Arno B. J. Kuijlaars ; Pablo Román
• 关键词：Orthogonal polynomials ; Riemann–Hilbert problems ; Asymptotic representations in the complex domain ; Limiting zero distribution ; Bessel functions ; 33C47 ; 34M50 ; 30E15 ; 33C10
• 刊名：Constructive Approximation
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：43
• 期：1
• 页码：153-196
• 全文大小：935 KB
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  6.Deift, P., Its, A., Krasovsky, I.: Asymptotics of Toeplitz, Hankel, and Toeplitz\(+\) Hankel determinants with Fisher–Hartwig singularities. Ann. Math. 174, 1243–1299 (2011)MATH MathSciNet CrossRef
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• 作者单位：Alfredo Deaño (1)
  Arno B. J. Kuijlaars (2)
  Pablo Román (3)
  
  1. Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911, Leganés, Madrid, Spain
  2. Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3001, Leuven, Belgium
  3. CIEM, FaMAF, Universidad Nacional de Córdoba, Medina Allende s/n Ciudad Universitaria, Córdoba, Argentina
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Analysis
  
• 出版者：Springer New York
• ISSN：1432-0940

文摘

We consider polynomials \(P_n\) orthogonal with respect to the weight \(J_{\nu }\) on \([0,\infty )\), where \(J_{\nu }\) is the Bessel function of order \(\nu \). Asheim and Huybrechs considered these polynomials in connection with complex Gaussian quadrature for oscillatory integrals. They observed that the zeros of \(P_n\) are complex and accumulate as \(n \rightarrow \infty \) near the vertical line \({{\mathrm{{\text {Re}}\,}}}z = \frac{\nu \pi }{2}\). We prove this fact for the case \(0 \le \nu \le 1/2\) from strong asymptotic formulas that we derive for the polynomials \(P_n\) in the complex plane. Our main tool is the Riemann–Hilbert problem for orthogonal polynomials, suitably modified to cover the present situation, and the Deift–Zhou steepest descent method. A major part of the work is devoted to the construction of a local parametrix at the origin, for which we give an existence proof that only works for \(\nu \le 1/2\). Keywords Orthogonal polynomials Riemann–Hilbert problems Asymptotic representations in the complex domain Limiting zero distribution Bessel functions