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 DankaEmail author


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.


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

Mathematics Subject Classification

42C05 33C10 33C45 


  1. 1.
    Deaño, A., Gil, A., Segura, J.: New inequalities from classical Sturm theorems. J. Approx. Theory 131, 208–230 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 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)Google Scholar
  3. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (2005)CrossRefGoogle Scholar
  5. 5.
    Ismail, M.E.H., Muldoon, M.E.: Zeros of combinations of Bessel functions and their derivatives. Appl. Anal. 21, 73–90 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kuijlaars, A.B.J., Vanlessen, M.: Universality for eigenvalue correlations at the origin of the spectrum. Commun. Math. Phys. 243, 163–191 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lubinsky, D.S.: A new approach to universality limits involving orthogonal polynomials. Ann. Math. 170, 915–939 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mastroianni, G., Totik, V.: Uniform spacing of zeros of orthogonal polynomials. Constr. Approx. 32, 181–192 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Minakshisundaram, S., Szász, O.: On absolute convergence of multiple Fourier series. Trans. Am. Math. Soc. 61, 36–53 (1947)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ransford, T.: Potential Theory in the Complex Plane. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  14. 14.
    Simon, B.: Two extensions of Lubinsky’s universality theorem. J. d’Analyse Math. 105, 345–362 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Stahl, H., Totik, V.: General Orthogonal Polynomials. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1992)CrossRefzbMATHGoogle Scholar
  16. 16.
    Totik, V.: Universality and fine zero spacing on general sets. Ark. Mat. 47, 361–391 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Varga, T.: Uniform spacing of zeros of orthogonal polynomials for locally doubling measures. Analysis (Munich) 33, 1–12 (2013)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1922)zbMATHGoogle Scholar

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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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