# zbMATH — the first resource for mathematics

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: