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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On a generalized moment problem. II


Authors: J. S. Hwang and G. D. Lin
Journal: Proc. Amer. Math. Soc. 91 (1984), 577-580
MSC: Primary 44A60; Secondary 26A48
DOI: https://doi.org/10.1090/S0002-9939-1984-0746093-4
MathSciNet review: 746093
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Abstract: Recently, we have extended the well-known Müntz-Szász theorem by showing that if $ f(z)$ is absolutely continuous and $ \vert f'(x)\vert \geqslant k > 0$ a.e. on $ (a,b)$, where $ a \geqslant 0$ and if $ \{ {n_p}\} $ is a sequence of positive numbers tending to infinity and satisfying $ \sum _{p = 1}^\infty 1/{n_p} = \infty $, then the sequence $ \{ f{(x)^{{n_p}}}\} $ is complete on $ (a,b)$ if and only if $ f(x)$ is strictly monotone on $ (a,b)$. We now apply Zarecki's theorem to improve the condition " $ \vert f'(x)\vert \geqslant k > 0$ a.e. on $ (a,b)$" by the condition $ f'(x) \ne 0$ a.e. on $ (a,b)$". Furthermore, we extend some well-known theorems of Picone, Mikusiński, and Boas.


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DOI: https://doi.org/10.1090/S0002-9939-1984-0746093-4
Keywords: Completeness, moment problem, absolutely continuous and monotone function
Article copyright: © Copyright 1984 American Mathematical Society