Jump distributions on $[-1,1]$ whose orthogonal polynomials have leading coefficients with given asymptotic behavior

Abstract:

We construct an explicit even jump distribution $d\alpha (x)$ on $[-1,1]$ for which ${\gamma _n}(d\alpha )$, the leading coefficient of the $n$th orthonormal polynomial for $d\alpha (x)$, has any given asymptotic behavior. For example, if $\{ {\xi _n}\} _1^\infty$ is a strictly increasing sequence with limit $\infty$, and ${\xi _{{3^{n + 1}}}}/{\xi _{{3^n}}} \to 1$ as $n \to \infty$, we can ensure that ${\lim _{n \to \infty }}{\gamma _n}(d\alpha ){2^{ - n}}/{\xi _n} = 1$. As a consequence, we have ${\lim _{n \to \infty }}{\gamma _{n - 1}}(d\alpha )/{\gamma _n}(d\alpha ) = 1/2$; that is, $d\alpha$ belongs to Nevai’s class $\mathcal {M}$. This positively and explicitly answers a question of Al. Magnus.