 03341 Barry Simon
 Ratio asymptotics and weak asymptotic measures for orthogonal polynomials on the real line
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Jul 21, 03

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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 $p_n^2 \, d\mu$
($p_n =P_n/\P_n\$) as $n\to\infty$ for orthogonal polynomials on the real line. We show existence
of ratio asymptotics at a single $z_0$ with $\Ima (z_0)\neq 0$ implies $d\mu$ is in a Nevai
class (i.e., $a_n\to a$ and $b_n \to b$ where $a_n,b_n$ are the offdiagonal and diagonal Jacobi
parameters). For $\mu$'s with bounded support, we prove $p_n^2\, d\mu$ 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.
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