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.