Composing functions of bounded 𝜑-variation

Ciemnoczołowski, J.; Orlicz, W.

doi:10.1090/S0002-9939-1986-0822434-6

Composing functions of bounded $\varphi$-variation
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by J. Ciemnoczołowski and W. Orlicz PDF

Proc. Amer. Math. Soc. 96 (1986), 431-436 Request permission

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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