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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 \(L_ p({\mathbb{R}})\).

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