Singularly continuous measures in Nevai’s class 𝑀

Abstract

Let d ν d\nu be a nonnegative Borel measure on [ − π , π ] [ - \pi ,\pi ] , with 0 > ∫ − π π d ν > ∞ 0 > \smallint _{ - \pi }^\pi d\nu > \infty and with support of Lebesgue measure zero. We show that there exist { η j } j = 1 ∞ ⊂ ( 0 , ∞ ) \{ {\eta _j}\} _{j = 1}^\infty \subset (0,\infty ) and { t j } j = 1 ∞ ⊂ ( − π , π ) \{ {t_j}\} _{j = 1}^\infty \subset ( - \pi ,\pi ) such that if \[ d μ ( θ ) := ∑ j = 1 ∞ η j d ν ( θ + t j ) , θ ∈ [ − π , π ] , d\mu (\theta ): = \sum \limits _{j = 1}^\infty {{\eta _j}d\nu (\theta + {t_j}),\quad \theta \in [ - \pi ,\pi ],} \] (with the usual periodic extension d ν ( θ ± 2 π ) = d ν ( θ ) d\nu (\theta \pm 2\pi ) = d\nu (\theta ) ), then the leading coefficients { κ n ( d μ ) } n = 0 ∞ \{ {\kappa _n}(d\mu )\} _{n = 0}^\infty of the orthonormal polynomials for d μ d\mu satisfy \[ lim n → ∞ κ n ( d μ ) / κ n + 1 ( d μ ) = 1. \lim \limits _{n \to \infty } {\kappa _n}(d\mu )/{\kappa _{n + 1}}(d\mu ) = 1. \] As a consequence, we obtain pure singularly continuous measures d α d\alpha on [ − 1 , 1 ] [ - 1,1] lying in Nevai’s class M M .