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Proceedings of the American Mathematical Society

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

 

 

On a generalized moment problem


Author: J. S. Hwang
Journal: Proc. Amer. Math. Soc. 87 (1983), 88-89
MSC: Primary 44A60
MathSciNet review: 677238
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Abstract: The well-known Müntz-Szász theorem asserts that the sequence of powers $ {x^{{n_p}}}$ is complete on $ [a,b]$, where $ a \geqslant 0$, if and only if (1)

$\displaystyle (1)\quad \sum\limits_{p = 1}^\infty {\frac{1} {{{n_p}}} = \infty ,\quad {\text{where}}\;0 < {n_1} < {n_2} < \cdots .} $

Let $ f(x)$ be absolutely continuous, $ \left\vert {f'(x)} \right\vert \geqslant k > 0$, and $ f(a)f(b) \geqslant 0$. We prove that under the assumption (1) the sequence $ \left\{ {f{{(x)}^{{n_p}}}} \right\}$ is complete on $ [a,b]$ if and only if $ f(x)$ is monotone on $ [a,b]$.

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DOI: https://doi.org/10.1090/S0002-9939-1983-0677238-1
Keywords: Completeness, moment problem, absolutely continuous and monotone function
Article copyright: © Copyright 1983 American Mathematical Society