Ratio asymptotics and weak asymptotic measures for orthogonal polynomials on the real line

Abstract

We study ratio asymptotics, that is, existence of the limit of Pn_(+1)(z)/P_n(z) (P_n= monic orthogonal polynomial) and the existence of weak limits of pn^2dμ(p_n=P_n/||P_n||) as n→∞ for orthogonal polynomials on the real line. We show existence of ratio asymptotics at a single z_0 with Im(z_0)≠0 implies dμ is in a Nevai class (i.e., a_n→a and bn→b where a_n,b_n are the off-diagonal and diagonal Jacobi parameters). For μ's with bounded support, we prove pn^2dμ has a weak limit if and only if limb_n, lima_(2n), and lima_(2n+1) all exist. In both cases, we write down the limits explicitly.

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