Freud's conjecture and approximation of reciprocals of weights by polynomials

LetW(x) be a function that is nonnegative inR, positive on a set of positive measure, and such that all power moments ofW ² (x) are finite. Let {p _n (W ²;x)} ₀ ^∞ denote the sequence of orthonormal polynomials with respect to the weightW ², and let {α _n } ₁ ^∞ and {β _n } ₁ ^∞ denote the coefficients in the recurrence relation $$xp_n (W^2 ,x) = alpha _{n + 1} p_{n + 1} (W^2 ,x) + beta _n p_n (W^2 ,x) + alpha _n p_{n - 1} (W^2 ,x).$$ We obtain a sufficient condition, involving mean approximation ofW ^?1 by reciprocals of polynomials, for $$mathop {lim }limits_{n to infty } {{alpha _n } mathord{left/ {vphantom {{alpha _n } {c_n }}} right. kern-nulldelimiterspace} {c_n }} = tfrac{1}{2}andmathop {lim }limits_{n to infty } {{beta _n } mathord{left/ {vphantom {{beta _n } {c_{n + 1} }}} right. kern-nulldelimiterspace} {c_{n + 1} }} = 0,$$ wherec _{n 1} ^∞ is a certain increasing sequence of positive numbers. In particular, we obtain a sufficient condition for Freud's conjecture associated with weights onR.