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Singularly continuous measures in Nevai's class $ M$


Author: D. S. Lubinsky
Journal: Proc. Amer. Math. Soc. 111 (1991), 413-420
MSC: Primary 42C05; Secondary 39A10
DOI: https://doi.org/10.1090/S0002-9939-1991-1039259-3
MathSciNet review: 1039259
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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

$\displaystyle 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

$\displaystyle \mathop {\lim }\limits_{n \to \infty } {\kappa _n}(d\mu )/{\kappa _{n + 1}}(d\mu ) = 1.$

As a consequence, we obtain pure singularly continuous measures $ d\alpha $ on $ [ - 1,1]$ lying in Nevai's class $ M$.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1039259-3
Keywords: Orthogonal polynomials, recurrence relations, Nevai's class $ M$, singularly continuous measures
Article copyright: © Copyright 1991 American Mathematical Society

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