On an Asymptotic Equality for Reproducing Kernels and Sums of Squares of Orthonormal Polynomials

  • A. Ignjatovic
  • D. S. Lubinsky
Part of the Springer Optimization and Its Applications book series (SOIA, volume 117)


In a recent paper, the first author considered orthonormal polynomials \(\left \{p_{n}\right \}\) associated with a symmetric measure with unbounded support and with recurrence relation
$$\displaystyle{ xp_{n}\left (x\right ) = A_{n}p_{n+1}\left (x\right ) + A_{n-1}p_{n-1}\left (x\right ),\quad n \geq 0. }$$
Under appropriate restrictions on \(\left \{A_{n}\right \}\), the first author established the identity
$$\displaystyle{ \lim _{n\rightarrow \infty }\frac{\sum _{k=0}^{n}p_{k}^{2}\left (x\right )} {\sum _{k=0}^{n}A_{k}^{-1}} =\lim _{n\rightarrow \infty }\frac{p_{2n}^{2}\left (x\right ) + p_{2n+1}^{2}\left (x\right )} {A_{2n}^{-1} + A_{2n+1}^{-1}}, }$$
uniformly for x in compact subsets of the real line. Here, we establish and evaluate this limit for a class of even exponential weights, and also investigate analogues for weights on a finite interval, and for some non-even weights.


Orthogonal polynomials Christoffel functions Recurrence coefficients 

2000 Mathematics Subject Classification




The research of second author supported by NSF grant DMS136208.


  1. 1.
    Badkov, V.M.: The Asymptotic Behavior of Orthogonal Polynomials. Math. USSR Sbornik 37, 39–51 (1980)CrossRefMATHGoogle Scholar
  2. 2.
    Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: A Riemann-Hilbert Approach to Asymptotic Questions for Orthogonal Polynomials. J. Comput. Appl. Math. 133, 47–63 (2001)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Freud, G.: Orthogonal Polynomials. Pergamon Press/Akademiai Kiado, Budapest (1971)MATHGoogle Scholar
  4. 4.
    Ignjatovic, A.: Asymptotic Behavior of Some Families of Orthonormal Polynomials and an Associated Hilbert Space. J. Approx. Theory 210, 41–79 (2016)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Levin, E., Lubinsky, D.S.: Orthogonal Polynomials for Exponential Weights. Springer, New York (2001)CrossRefMATHGoogle Scholar
  6. 6.
    Mate, A., Nevai, P., Totik, V.: Extensions of Szegő’s Theory of Orthogonal Polynomials, III. Constr. Approx. 3, 73–96 (1987)CrossRefMATHGoogle Scholar
  7. 7.
    Mate, A., Nevai, P., Totik, V.: Szegő’s Extremum Problem on the Unit Circle. Ann. Math. 134, 433–453 (1991)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Mhaskar, H.N., Saff, E.B.: Extremal Problems for Polynomials with Exponential Weights. Trans. Am. Math. Soc. 285, 203–234 (1984)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Mhaskar, H.N., Saff, E.B.: Where does the L p norm of a weighted polynomial live?. Trans. Am. Math. Soc. 303, 109–124 (1987)MathSciNetMATHGoogle Scholar
  10. 10.
    Simon, B.: Szegő’s Theorem and Its Descendants. Princeton University Press, Princeton (2011)MATHGoogle Scholar
  11. 11.
    Szegő, G.: Orthogonal Polynomials. American Mathematical Society, Providence (1939)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.School of Computer Science and EngineeringUniversity of New South WalesSydneyAustralia
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations