Zeros of orthogonal polynomials on the real line

Abstract

Let $p_n(x)$ be orthogonal polynomials associated to a measure $d\mu$ of compact support in $R$. If $E\not\in supp(d\mu)$, we show there is a $\delta>0$ so that for all $n$, either $p_n$ or $p_{n+1}$ has no zeros in $(E-\delta, E+\delta)$. If $E$ is an isolated point of $supp(d\mu)$, we show there is a $\delta$ so that for all $n$, either $p_n$ or $p_{n+1}$ has at most one zero in $(E-\delta, E+\delta)$. We provide an example where the zeros of $p_n$ are dense in a gap of $supp(d\mu)$.