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 P n+1 ( z)/ P n ( z) ( P n = monic orthogonal polynomial) and the existence of weak limits of p n 2 dμ (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 b n → b where a n , b n are the off-diagonal and diagonal Jacobi parameters). For μ's with bounded support, we prove p n 2 dμ has a weak limit if and only if lim b n , lim a 2n , and lim a 2n+1 all exist. In both cases, we write down the limits explicitly.