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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Composing functions of bounded $ \varphi$-variation


Authors: J. Ciemnoczołowski and W. Orlicz
Journal: Proc. Amer. Math. Soc. 96 (1986), 431-436
MSC: Primary 26A45; Secondary 26A16
MathSciNet review: 822434
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Abstract: Let $ {F_n}$ be finite-valued functions on $ ( - \infty ,\infty ),{\text{ }}{F_n}(0) = 0,{\text{ }}n = 1,2, \ldots .$. For $ x \in {\mathcal{V}_\varphi }\left\langle {a,b} \right\rangle $, the class of functions of bounded $ \varphi $-variation, the compositions $ {F_n}(x)$ are studied. The main result of this paper is Theorem 1 stating necessary and sufficient conditions for the sequence $ {\operatorname{var} _\psi }({F_n}(x),a,b)$ to be bounded for each $ x \in {\mathcal{V}_\varphi }\left\langle {a,b} \right\rangle $ ($ \psi $ denotes here another $ \varphi $-function).


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DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0822434-6
PII: S 0002-9939(1986)0822434-6
Keywords: Function of bounded $ \varphi $-variation, composition, Lipschitz condition
Article copyright: © Copyright 1986 American Mathematical Society