## A proof of Freud’s conjecture for exponential weights.(English)Zbl 0653.42024

Let W(x) be a function nonnegative in $${\mathbb{R}}$$, positive an a set of positive measure, and such that all power moments of $$W^ 2(x)$$ are finite. Let $$\{p_ n(W^ 2,x)\}^{\infty}_ 0$$ denote the sequence of orthonormal polynomials with respect to the weight $$W^ 2(x)$$, and let $$\{A_ n\}^{\infty}_ 1$$ and $$\{B_ n\}^{\infty}_ 1$$ denote the coefficients in the recurrence relation $xp_ n(W^ 2,x)=A_{n+1}p_{n+1}(W^ 2,x)+B_ np_ n(W^ 2,x)+A_ np_{n- 1}(W^ 2,x).$ When $$W(x)=w(x)\exp (-Q(x))$$, $$x\in (-\infty,\infty)$$, where w(x) is a “generalized Jacobi factor”, and Q(x) satisfies various restrictions, we show that $$\lim_{n\to \infty}A_ n/a_ n=$$ and $$\lim_{n\to \infty}B_ n/a_ n=0,$$ where, for n large enough, $$a_ n$$ is a positive root of the equation $n=(2/\pi)\int^{1}_{0}a_ nxQ'(a_ nx)(1-x^ 2)^{-1/2} dx.$ In the special case, $$Q(x)=| x|^{\alpha}$$, $$\alpha >0$$, this proves a conjecture due to G. Freud. We also consider various noneven weights, and establish certain infinite-finite range inequalities for weighted polynomials in $$L_ p({\mathbb{R}})$$.

### MSC:

 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 41A25 Rate of convergence, degree of approximation

### Keywords:

power moments; weight
Full Text:

### References:

  L. Ahlfors (1953): Complex Analysis. New York: McGraw-Hill. · Zbl 0052.07002  W. C. Bauldry, A. Máté, P. Nevai (to appear):Asymptotics for the solutions of systems of smooth recurrence equations. Pacific J. Math. · Zbl 0663.42024  D. Bessis, C. Itzykson, J. B. Zuber (1980):Quantum field theory techniques in graphical enumeration. Adv. in Appl. Math.,1:109–157. · Zbl 0453.05035  H. B. Dwight (1961): Tables of Integrals and Other Mathematical Data, 4th edn. New York: Macmillan. · Zbl 0154.18410  G. Freud (1971): Orthogonal Polynomials. Budapest: Akademiai Kiado-Pergamon Press. · Zbl 0226.33014  G. Freud (1976):On the coefficients in the recursion formulae of orthogonal polynomials. Proc. Roy. Irish Acad. Sect. A,76:1–6. · Zbl 0327.33008  A. Knopfmacher, D. S. Lubinsky, P. Nevai (1988):Freud’s conjecture and approximation of reciprocals of weights by polynomials. Constr. Approx.,4:9–20. · Zbl 0646.41002  D. S. Lubinsky (unpublished): Variation on a Theme of Mhaskar, Rahmanov, and Saff, or ”Sharp” Weighted Polynomial Inequalities inLp(R). NRIMS Internal Report No. 1575, Pretoria, 1984.  D. S. Lubinsky (1985):Even entire functions absolutely monotone in [0, and weights on the whole real line. In: Orthogonal Polynomials and Their Applications (C. Brezinski,et al., eds.) Lecture Notes in Mathematics, vol. 1171. Berlin: Springer-Verlag. · Zbl 0599.41048  D. S. Lubinsky (1986):Gaussian quadrature, weights on the whole real line and even entire functions with nonnegative even order derivatives. J. Approx. Theory,46:297–313. · Zbl 0608.41017  D. S. Lubinsky, E. B. Saff (1988):Uniform and mean approximation by certain weighted polynomials, with applications. Constr. Approx.,4:21–64. · Zbl 0646.41003  D. S. Lubinsky, H. N. Mhaskar, E. B. Saff (1986):Freud’s conjecture for exponential weights. Bull. Amer. Math. Soc.,15:217–221. · Zbl 0606.42018  Al. Magnus (1985):A proof of Freud’s conjecture about orthogonal polynomials related to |x| p exp(-x 2m ). In: Orthogonal Polynomials and Their Applications (C. Brezinskiet al., eds.) Lecture Notes in Mathematics, vol. 1171. Berlin: Springer-Verlag.  Al. Magnus (1986):On Freud’s equations for exponential weights. J. Approx. Theory,46:65–99. · Zbl 0619.42015  A. Máté, P. Nevai, V. Totik (1985):Asymptotics for the ratio of leading coefficients of orthogonal polynomials on the unit circle. Constr. Approx.,1:63–69. · Zbl 0582.42012  A. Máté, P. Nevai, T. Zaslavsky (1985):Asymptotic expansion of ratios of coefficients of orthogonal polynomials with exponential weights. Trans. Amer. Math. Soc.,287:495–505. · Zbl 0536.42023  H. N. Mhaskar, E. B. Saff (1984):Extremal problems for polynomials with exponential weights. Trans. Amer. Math. Soc.,285:203–234. · Zbl 0546.41014  H. N. Mhaskar, E. B. Saff (1984):Weighted polynomials on finite and infinite intervals: a unified approach. Bull. Amer. Math. Soc.,11:351–354. · Zbl 0565.41017  H. N. Mhaskar, E. B. Saff (1985):Where does the sup norm of a weighted polynomial live? (A generalization of incomplete polynomials). Constr. Approx.,1:71–91. · Zbl 0582.41009  H. N. Mhaskar, E. B. Saff (to appear):Where does the L p norm of a weighted polynomial live? Trans. Amer. Math. Soc.  P. Nevai (1979):Orthogonal polynomials. Mem. Amer. Math. Soc.,213:1–185. · Zbl 0405.33009  P. Nevai (1986):Geza Freud, Christoffel functions and orthogonal polynomials (A case study). J. Approx. Theory,48:3–167. · Zbl 0606.42020  P. Nevai, V. Totik (1986):Sharp Nikolskii-type estimates for exponential weights. Constr. Approx.,2:113–127. · Zbl 0604.41014  D. G. Pettifor, D. L. Weaire, eds. (1984): The Recursion Method and Its Applications. Series in Solid-State Physics, vol. 58. New York: Springer-Verlag. · Zbl 0614.00025  E. A. Rahmanov (1983):On the asymptotics of the ratio of orthogonal polynomials, II. Math. USSR-Sb.,46:105–117. · Zbl 0515.30030  E. A. Rahmanov (1984):Asymptotic properties of orthogonal polynomials on the real axis. Math. USSR-Sb.,47:155–193. · Zbl 0522.42018  E. B. Saff (1983):Incomplete and orthogonal polynomials. In: Approximation Theory IV. (C. K. Chui, L. L. Schumaker, J. D. Ward, eds.) New York: Academic Press, pp. 219–256. · Zbl 0563.41006
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