Open Access
2008/2009 On Functions of (p,α)-Bounded Variation
René Erlín Castillo, Eduard Trousselot
Real Anal. Exchange 34(1): 49-60 (2008/2009).


In this paper we introduce the concept of $\;(p,\alpha)$-bounded variation which generalizes the Riesz p-variation. The following result is proved: a function $\; f:[a,b] \rightarrow \mathbb{R}\;$ is of $\; (p,\alpha)$-bounded variation $\;(1<p<\infty)\;$ if and only if $f$ is $\alpha$-absolutely continuous on $[a,b]$ and $f^{^{\prime}}_{\alpha} \in L_{(p,\alpha)} [a,b]$. Moreover it is shown that the $\;(p,\alpha)$-bounded variation of a function $f$ on $[a,b]$ is given by \begin{equation*} V_{(p,\alpha)} (f) = \|f^{^{\prime}}_{\alpha}\|^{p}_{L_{(p,\alpha)} [a,b]}. \end{equation*}


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René Erlín Castillo. Eduard Trousselot. "On Functions of (p,α)-Bounded Variation." Real Anal. Exchange 34 (1) 49 - 60, 2008/2009.


Published: 2008/2009
First available in Project Euclid: 19 May 2009

MathSciNet: MR2527121

Primary: 26A45 , 26B30
Secondary: 26A16 , 26A24

Keywords: (p,α)-bounded variation , Riesz p-variation

Rights: Copyright © 2008 Michigan State University Press

Vol.34 • No. 1 • 2008/2009
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