Singularly continuous measures in Nevai’s class $M$

Abstract:

Let $d\nu$ be a nonnegative Borel measure on $[ - \pi ,\pi ]$, with $0 < \smallint _{ - \pi }^\pi d\nu < \infty$ and with support of Lebesgue measure zero. We show that there exist $\{ {\eta _j}\} _{j = 1}^\infty \subset (0,\infty )$ and $\{ {t_j}\} _{j = 1}^\infty \subset ( - \pi ,\pi )$ such that if \[ d\mu (\theta ): = \sum \limits _{j = 1}^\infty {{\eta _j}d\nu (\theta + {t_j}),\quad \theta \in [ - \pi ,\pi ],} \](with the usual periodic extension $d\nu (\theta \pm 2\pi ) = d\nu (\theta )$), then the leading coefficients $\{ {\kappa _n}(d\mu )\} _{n = 0}^\infty$ of the orthonormal polynomials for $d\mu$ satisfy \[ \lim \limits _{n \to \infty } {\kappa _n}(d\mu )/{\kappa _{n + 1}}(d\mu ) = 1.\] As a consequence, we obtain pure singularly continuous measures $d\alpha$ on $[ - 1,1]$ lying in Nevai’s class $M$.