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Constructive Approximation

, Volume 47, Issue 3, pp 407–435 | Cite as

Zeros of Orthogonal Polynomials Near an Algebraic Singularity of the Measure

  • Árpád Baricz
  • Tivadar Danka
Article
  • 121 Downloads

Abstract

In this paper, we study the local zero behavior of orthogonal polynomials around an algebraic singularity, that is, when the measure of orthogonality is supported on \( [-1,1] \) and behaves like \( h(x)|x - x_0|^\lambda dx \) for some \( x_0 \in (-1,1) \), where h(x) is strictly positive and analytic. We shall sharpen the theorem of Yoram Last and Barry Simon and show that the so-called fine zero spacing (which is known for \( \lambda = 0\)) unravels in the general case, and the asymptotic behavior of neighbouring zeros around the singularity can be described with the zeros of the function \( c J_{\frac{\lambda - 1}{2}}(x) + d J_{\frac{\lambda + 1}{2}}(x) \), where \( J_a(x) \) denotes the Bessel function of the first kind and order a. Moreover, using Sturm–Liouville theory, we study the behavior of this linear combination of Bessel functions, thus providing estimates for the zeros in question.

Keywords

Orthogonal polynomials Fine zero spacing Generalized Jacobi measure Bessel function Riemann–Hilbert method 

Mathematics Subject Classification

42C05 33C10 33C45 

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Authors and Affiliations

  1. 1.Department of EconomicsBabeş-Bolyai UniversityCluj-NapocaRomania
  2. 2.Institute of Applied MathematicsÓbuda UniversityBudapestHungary
  3. 3.Bolyai InstituteUniversity of SzegedSzegedHungary

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