Opuscula Math. 32, no. 1 (2012), 137-151
http://dx.doi.org/10.7494/OpMath.2012.32.1.137

Opuscula Mathematica

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

José Giménez
Nelson Merentes
Sergio Rivas

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.

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• José Giménez
• Universidad de Los Andes, Departamento de Matemáticas, Facultad de Ciencias, Mérida, Venezuela
• Nelson Merentes
• Universidad Central de Venezuela, Escuela de Matemáticas, Caracas, Venezuela
• Sergio Rivas
• Universidad Nacional Abierta, Departamento de Matemáticas, Caracas, Venezuela
• Revised: 2011-11-28.
• Accepted: 2011-11-30.