03-341 Barry Simon

Ratio asymptotics and weak asymptotic measures for orthogonal polynomials on the real line (48K, AMS-LaTeX) 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 off-diagonal 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.

Files: 03-341.src( 03-341.keywords , simon_ratios.TEX )