Constructive Approximation

, Volume 8, Issue 4, pp 463–535 | Cite as

Christoffel functions, orthogonal polynomials, and Nevai's conjecture for Freud weights

  • A. L. Levin
  • D. S. Lubinsky


We obtain upper and lower bounds for Christoffel functions for Freud weights by relatively new methods, including a new way to estimate discretization of potentials. We then deduce bounds for orthogonal polynomials onℝ thereby largely resolving a 1976 conjecture of P. Nevai. For example, let W:=e−Q, whereQ:ℝ→ℝ is even and continuous inℝ, Q" is continuous in (0, ∞) andQ'>0 in (0, ∞), while, for someA, B,
$$1< A \leqslant \frac{{(d/dx)(xQ'(x))}}{{Q'(x)}} \leqslant B,x \in (0,\infty )$$
Letan denote thenth Mhaskar-Rahmanov-Saff number forQ, andL>0. Then, uniformly forn≥1 and |x|≤an(1+Ln−2/3),
$$\lambda _n (W^2 ,x) \sim \frac{{a_n }}{n}W^2 (x)\left( {\max \left\{ {n^{ - 2/3} ,1 - \frac{{|x|}}{{a_n }}} \right\}} \right)^{ - 1/2}$$

Moreover, for all x εℝ, we can replace ∼ by ≥. In particular, these results apply toW(x):=exp(-|x|α), α>1. We also obtain lower bounds for allx εℝ, when onlyA>0, but this necessarily requires a more complicated formulation.

We deduce that thenth orthonormal plynomialpn(W2, ·). forW2 satisfies
$$\mathop {\sup }\limits_{x \in \mathbb{R}} |p_n (W^2 ,x)|W(x)\left| {1 - \frac{{|x|}}{{a_n }}} \right|^{1/4} \sim a_n^{ - 1/2}$$
$$\mathop {\sup }\limits_{x \in \mathbb{R}} |p_n (W^2 ,x)|W(x) \sim a_n^{ - 1/2} n^{1/6} .$$

In particular, this applies toW(x):=exp(-|x|α), α>1.

AMS classification

Primary 41A17 42C05 Secondary 41A10 

Key words and phrases

Freud weights Exponential weights Orthonormal polynomials Christoffel functions Markov-Bernstein inequalities Potentials Discretization of potentials Nevai's conjecture 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    W. C. Bauldry (1990):Estimates of asymmetric Freud polynomials on the real line. J. Approx. Theory,63:225–237.Google Scholar
  2. 2.
    S. S. Bonan (1983):Applications of G. Freud's theory I. In: Approximation Theory IV (C. K. Chui, L. L. Schumaker, J. D. Ward, eds.). New York: Academic Press, pp. 347–351.Google Scholar
  3. 3.
    S. S. Bonan, D. S. Clark (1990):Estimates of the Hermite and the Freud polynomials. J. Approx. Theory,63:210–224.Google Scholar
  4. 4.
    J. Clunie, T. Kövari (1968):On integral functions having prescribed asymptotic growth II. Canad. J. Math.,20:7–20.Google Scholar
  5. 5.
    G. Freud (1977):On estimations of the greatest zeros of orthogonal polynomials. Acta Math. Acad. Sci. Hungar.,25:99–107.Google Scholar
  6. 6.
    G. Freud (1977):On Markov-Bernstein type inequalities and their applications. J. Approx. Theory,19:22–37.Google Scholar
  7. 7.
    G. Freud, A. Giroux, Q. I. Rahman (1978):On approximation by polynomials with weight exp(-|x|). Canad. J. Math.,30:358–372 (in French).Google Scholar
  8. 8.
    T. Ganelius (1976):Rational approximation in the complex plane and on the line. Ann. Acad. Sci. Fenn.,2:129–145.Google Scholar
  9. 9.
    A. Knopfmacher, D. S. Lubinsky (1987):Mean convergence of Lagrange interpolation for Freud's weights with application to product integration rules. J. Comput. Appl. Math.,17:79–103.Google Scholar
  10. 10.
    A. L. Levin, D. S. Lubinsky (1987):Canonical products and the weights exp(-|x|α), α>1,with applications. J. Approx. Theory,49:149–169.Google Scholar
  11. 11.
    A. L. Levin, D. S. Lubinsky (1987):Weights on the real line that admit good relative polynomial approximation, with applications. J. Approx. Theory.,49:170–195.Google Scholar
  12. 12.
    A. L. Levin, D. S. Lubinsky (1990):L Markov and Bernstein inequalities for Freud weights. SIAM J. Math. Anal.,21:1065–1082.Google Scholar
  13. 13.
    D. S. Lubinsky (1986):Gaussian quadrature, weights on the whole real line, and even entive functions with non-negative even order derivatives. J. Approx. Theory,46:297–313.Google Scholar
  14. 14.
    D. S. Lubinsky (1989): Strong Asymptotics for Extremal Errors and Polynomials Associated with Erdös-Type Weights. Pitman Research Notes in Mathematics, vol. 202. Harlow, Essex: Longman.Google Scholar
  15. 15.
    D. S. Lubinsky, E. B. Saff (1988): Strong Asymptotics for Extremal polynomials Associated with Exponential Weights. Lecture Notes in Mathematics, vol. 1305. Berlin: Springer-Verlag.Google Scholar
  16. 16.
    A. Mate, P. Nevai, V. Totik (1986):Asymptotics for the zeros of orthogonal polynomials associated with infinite intervals. J. London Math. Soc.,33:303–310.Google Scholar
  17. 17.
    H. N. Mhaskar (1990):Bounds for certain Freud-type orthogonal polynomials. J. Approx. Theory,63:238–254.Google Scholar
  18. 18.
    H. N. Mhaskar, E. B. Saff (1984):Extremal Problems for Polynomials with Exponential Weights. Trans. Amer. Math. Soc.,285:203–234.Google Scholar
  19. 19.
    H. N. Mhaskar, E. B. Saff (1985):Where does the sup-norm of a weighted polynomial live? Constr. Approx.,1:71–91.Google Scholar
  20. 20.
    H. N. Mhaskar, E. B. Saff (1987):Where does the L p-norm of a weighted polynomial live? Trans. Amer. Math. Soc.,303:109–124.Google Scholar
  21. 21.
    P. Nevai (1976):Lagrange interpolation at the zeros of orthogonal polynomials. In: Approximation Theory II (G. G. Lorentz, C. K. Chui, and L. L. Schumaker, eds.). New York: Academic Press, pp. 163–201.Google Scholar
  22. 22.
    P. Nevai (1979): Orthogonal Polynomials. Memoirs of the American Mathematical Society, no. 213. Providence, RI: American Mathematical Society.Google Scholar
  23. 23.
    P. Nevai (1984):Asymptotics for orthogonal polynomials associated with exp(−x 4). SIAM J. Math. Anal.,15:1177–1187.Google Scholar
  24. 24.
    P. Nevai (1986):Geza Freud, orthogonal polynomials and Christoffel functions. A case study. J. Approx. Theory,48:3–167.Google Scholar
  25. 25.
    P. Nevai, V. Totik (1986):Weighted polynomial inequalities. Constr. Approx.,2:113–127.Google Scholar
  26. 26.
    D. J. Newman, A. R. Reddy (1977):Rational approximation to |x|/(1+x 2m)on (−∞, ∞). J. Approx. Theory,19:231–238.Google Scholar
  27. 27.
    E. A. Rahmanov (1984):On asymptotic properties of polynomials orthogonal on the real axis. Math. USSR-Sb.,47:155–193.Google Scholar
  28. 28.
    E. A. Rahmanov (1991):Strong asymptotics for orthogonal polynomials associated with exponential weights onℝ. Manuscript.Google Scholar
  29. 29.
    R. C. Sheen (1987):Plancherel-Rotach type asymptotics for orthogonal polynomials associated with exp(−x 6/6). J. Approx. Theory,50:232–293.Google Scholar
  30. 30.
    G. A. Szegö (1975). Orthogonal Polynomials, 4th edn. American Mathematical Society Colloquium Publications, vol. 23. Providence, RI: American Mathematical Society.Google Scholar

Copyright information

© Springer-Verlag New York Inc 1992

Authors and Affiliations

  • A. L. Levin
    • 1
  • D. S. Lubinsky
    • 2
  1. 1.Department of MathematicsThe Open University of IsraelTel AvivIsrael
  2. 2.Department of MathematicsUniversity of WitwatersrandRepublic of South Africa

Personalised recommendations