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


Author: D. S. Lubinsky
Journal: Proc. Amer. Math. Soc. 104 (1988), 516-524
MSC: Primary 42C05
DOI: https://doi.org/10.1090/S0002-9939-1988-0962822-2
MathSciNet review: 962822
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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.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0962822-2
Keywords: Orthogonal polynomials, jump distributions, asymptotics, Nevai's class, leading coefficients, Jacobi matrices, discrete scattering
Article copyright: © Copyright 1988 American Mathematical Society

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