Constructive Approximation

, Volume 4, Issue 1, pp 65–83 | Cite as

A proof of Freud's conjecture for exponential weights

  • D. S. Lubinsky
  • H. N. Mhaskar
  • E. B. Saff
Article
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Abstract

LetW(x) be a function nonnegative inR, positive on a set of positive measure, and such that all power moments ofW2(x) are finite. Let {p n (W2;x)} 0 ∞ denote the sequence of orthonormal polynomials with respect to the weightW2(x), and let {A n } 1 ∞ and {B n } 1 ∞ denote the coefficients in the recurrence relation
$$xp_n (W^2 ,x) = A_{n + 1} p_{n + 1} (W^2 ,x) + B_n p_n (W^2 ,x) + A_n p_{n - 1} (W^2 ,x).$$
. WhenW(x) =w(x) exp(-Q(x)), xε(-∞,∞), wherew(x) is a “generalized Jacobi factor,” andQ(x) satisfies various restrictions, we show that
$$\mathop {\lim }\limits_{n \to \infty } {{A_n } \mathord{\left/ {\vphantom {{A_n } {a_n }}} \right. \kern-\nulldelimiterspace} {a_n }} = \tfrac{1}{2}and\mathop {\lim }\limits_{n \to \infty } {{B_n } \mathord{\left/ {\vphantom {{B_n } {a_n }}} \right. \kern-\nulldelimiterspace} {a_n }} = 0,$$
where, forn large enough,a n is the positive root of the equation
$$n = ({2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi })\int_0^1 {a_n xQ'(a_n x)(1 - x^2 )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} dx.}$$
In the special case, Q(x) = ¦x¦α, a > 0, this proves a conjecture due to G. Freud. We also consider various noneven weights, and establish certain infinite-finite range inequalities for weighted polynomials inLp(R).

AMS classification

Primary 41A25 Primary 42C05 

Key words and phrases

Exponential weights Freud's conjecture Orthogonal polynomials Recurrence relation coefficients 
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Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • D. S. Lubinsky
    • 1
  • H. N. Mhaskar
    • 2
  • E. B. Saff
    • 3
  1. 1.National Research Institute for Mathematical SciencesC.S.I.R.PretoriaRepublic of South Africa
  2. 2.Department of MathematicsBowling Green State UniversityBowling GreenUSA
  3. 3.Institute for Constructive Mathematics Department of MathematicsUniversity of South FloridaTampaUSA

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