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.