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

Gimenez J., Merentes N., Rivas S.

Opuscula Mathematica

2012

Vol. 32, no. 1

137-151

In this article we introduce the concept of second Φ--variation in the sense of Schramm for normed-space valued functions defined on an interval [a; b] ⊂ R. To that end we combine the notion of second variation due to de la Vallée Poussin and the concept of φ-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 [formula] (where X is a reflexive Banach space) is of bounded second Φ-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.