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Zeros of Orthogonal Polynomials Near an Algebraic Singularity of the Measure

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

References

  1. Deaño, A., Gil, A., Segura, J.: New inequalities from classical Sturm theorems. J. Approx. Theory 131, 208–230 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Deift, P.: Orthogonal polynomials and random matrices: A Riemann–Hilbert approach, Courant Lecture Notes in Mathematics 3. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence (1999)

  3. Foulquié Moreno, A., Martínez-Finkelshtein, A., Sousa, V.L.: Asymptotics of orthogonal polynomials for a weight with a jump on \( [-1,1] \). Constr. Approx. 33, 219–263 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (2005)

    Book  Google Scholar 

  5. Ismail, M.E.H., Muldoon, M.E.: Zeros of combinations of Bessel functions and their derivatives. Appl. Anal. 21, 73–90 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kuijlaars, A.B.J., McLaughlin, K.T.-R., Van Assche, W., Vanlessen, M.: The Riemann–Hilbert approach to strong asymptotics for orthogonal polynomials on \( [-1,1] \). Adv. Math. 188, 337–398 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kuijlaars, A.B.J., Vanlessen, M.: Universality for eigenvalue correlations at the origin of the spectrum. Commun. Math. Phys. 243, 163–191 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Last, Y., Simon, B.: Fine structure of the zeros of orthogonal polynomials IV. A priori bounds and clock behavior. Commun. Pure Appl. Math. 61, 486–538 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Levin, E., Lubinsky, D.S.: Applications of universality limits to zeros and reproducing kernels of orthogonal polynomials. J. Approx. Theory 150, 69–95 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lubinsky, D.S.: A new approach to universality limits involving orthogonal polynomials. Ann. Math. 170, 915–939 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Mastroianni, G., Totik, V.: Uniform spacing of zeros of orthogonal polynomials. Constr. Approx. 32, 181–192 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Minakshisundaram, S., Szász, O.: On absolute convergence of multiple Fourier series. Trans. Am. Math. Soc. 61, 36–53 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ransford, T.: Potential Theory in the Complex Plane. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  14. Simon, B.: Two extensions of Lubinsky’s universality theorem. J. d’Analyse Math. 105, 345–362 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Stahl, H., Totik, V.: General Orthogonal Polynomials. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1992)

    Book  MATH  Google Scholar 

  16. Totik, V.: Universality and fine zero spacing on general sets. Ark. Mat. 47, 361–391 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Vanlessen, M.: Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight. J. Approx. Theory 125(2), 198–237 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Varga, T.: Uniform spacing of zeros of orthogonal polynomials for locally doubling measures. Analysis (Munich) 33, 1–12 (2013)

    MathSciNet  MATH  Google Scholar 

  19. Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1922)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Economics, Babeş-Bolyai University, 400591, Cluj-Napoca, Romania

    Árpád Baricz

  2. Institute of Applied Mathematics, Óbuda University, Budapest, 1034, Hungary

    Árpád Baricz

  3. Bolyai Institute, University of Szeged, Szeged, 6720, Hungary

    Tivadar Danka

Authors
  1. Árpád Baricz
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  2. Tivadar Danka
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Correspondence to Tivadar Danka.

Additional information

Communicated by Percy A. Deift.

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.

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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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