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Proceedings of the American Mathematical Society

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

 

 

Some properties of asymptotic functions and their applications


Author: Ling Yau Chan
Journal: Proc. Amer. Math. Soc. 72 (1978), 239-247
MSC: Primary 26A16; Secondary 42A16
DOI: https://doi.org/10.1090/S0002-9939-1978-0507315-5
MathSciNet review: 507315
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Abstract: In this paper we give complete characterizations, in terms of Dini numbers and integrals, of positive functions $ \Phi (u)$ defined in (0, $ \infty $) satisfying the conditions: (i) $ \Phi (u)/{u^a}$ is nondecreasing and (ii) $ \Phi (u)/{u^b}$ is nonincreasing. By applying these results we obtain necessary and sufficient conditions for power series and trigonometric series to satisfy a certain Lipschitz condition, which include some known results of R. P. Boas, Jr. [1]. We also give complete characterizations of positive functions $ \Phi (u)$ defined in $ ( - \infty ,\infty )$ satisfying the conditions: (i) $ \Phi (u)/{e^{au}}$ is nondecreasing and (ii) $ \Phi (u)/{e^{bu}}$ is nonincreasing.


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DOI: https://doi.org/10.1090/S0002-9939-1978-0507315-5
Keywords: Asymptotic functions, interpolation, trigonometric series, power series, Lipschitz condition
Article copyright: © Copyright 1978 American Mathematical Society