Zeros of Orthogonal Polynomials Near an Algebraic Singularity of the Measure

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.

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Notes

  1. 1.

    It is important to mention here that the reality of the zeros of \( \varphi _{a,c,d}(x) \) and the infinite product actually imply that the function \( x \mapsto \frac{2^a}{c}\Gamma (a+1)x^{-a-1}\varphi _{a,c,d}(x) \) belongs to the Laguerre–Pólya class of real entire functions, and hence satisfies the so-called Laguerre inequality. The decreasing property of the logarithmic derivative of \( \varphi _{a,c,d}(x) \) is a consequence of this inequality.

  2. 2.

    The author Á. Baricz is very grateful to Christoph Koutchan for deducing this differential equation with his Holonomic Functions Package.

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Correspondence to Tivadar Danka.

Additional information

The research of Árpád Baricz was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. The research of Tivadar Danka was supported by ERC Advanced Grant No. 267055 and by the ÚNKP-ÚNKP-16-3 New National Excellence Program of the Ministry of Human Capacities.

Communicated by Percy A. Deift.

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Baricz, Á., Danka, T. Zeros of Orthogonal Polynomials Near an Algebraic Singularity of the Measure. Constr Approx 47, 407–435 (2018). https://doi.org/10.1007/s00365-017-9411-5

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Keywords

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

Mathematics Subject Classification

  • 42C05
  • 33C10
  • 33C45