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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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Keywords: Function of bounded <IMG WIDTH="19" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$\varphi$">-variation, composition, Lipschitz condition
Article copyright: © Copyright 1986 American Mathematical Society