Opuscula Mathematica
Opuscula Math. 32, no. 1 (), 137-151
Opuscula Mathematica

Integral representation of functions of bounded second Φ-variation in the sense of Schramm

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.
Mathematics Subject Classification: 26B30, 26B35.
Cite this article as:
José Giménez, Nelson Merentes, Sergio Rivas, Integral representation of functions of bounded second Φ-variation in the sense of Schramm, Opuscula Math. 32, no. 1 (2012), 137-151, http://dx.doi.org/10.7494/OpMath.2012.32.1.137
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ISSN 1232−9274, e-ISSN 2300−6919, DOI http://dx.doi.org/10.7494/OpMath
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