TY  - JOUR
ID  - Giménez2012
LB  - Giménez2012
AU  - Giménez, José
AU  - Merentes, Nelson
AU  - Rivas, Sergio
TI  - Integral representation of functions of bounded second Φ-variation in the sense of Schramm
ST  - Integral representation of functions of bounded second Φ-variation in the sense of Schramm
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2012
VL  - 32
IS  - 1
SP  - 137
EP  - 151
DO  - http://dx.doi.org/10.7494/OpMath.2012.32.1.137
UR  - http://dx.doi.org/10.7494/OpMath.2012.32.1.137
AB  - 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.
KW  - Young function
KW  - \(\Phi\)-variation
KW  - second \(\Phi\)-variation of a function
SN  - 1232-9274
ER  -