@Article{Giménez2012,
author = "Giménez, José and Merentes, Nelson and Rivas, Sergio",
title = "Integral representation of functions of bounded second Φ-variation in the sense of Schramm",
journal = "Opuscula Math.",
year = "2012",
volume = "32",
number = "1",
pages = "137--151",
doi = "http://dx.doi.org/10.7494/OpMath.2012.32.1.137",
url = "http://dx.doi.org/10.7494/OpMath.2012.32.1.137",
abstract = "In this article we introduce the concept of second \(\Phi\)-variation in the sense of Schramm for normed-space valued functions defined on an interval \([a,b] \subset \mathbb{R}\). To that end we combine the notion of second variation due to de la Vallée Poussin and the concept of \(\varphi\)-variation in the sense of Schramm for real valued functions. In particular, when the normed space is complete we present a characterization of the functions of the introduced class by means of an integral representation. Indeed, we show that a function \(f \in \mathbb{X}^{[a,b]}\) (where \(\mathbb{X}\) is a reflexive Banach space) is of bounded second \(\Phi\)-variation in the sense of Schramm if and only if it can be expressed as the Bochner integral of a function of (first) bounded variation in the sense of Schramm.",
keywords = "Young function, \(\Phi\)-variation, second \(\Phi\)-variation of a function",
issn = "1232-9274"
}