Opuscula Math. 32, no. 1 (2012), 137-151

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
  • Received: 2010-08-16.
  • Revised: 2011-11-28.
  • Accepted: 2011-11-30.
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Cite this article as:
José Giménez, Nelson Merentes, Sergio Rivas, Integral representation of functions of bounded second Φ-variation in the sense of Schramm, Opuscula Math. 32, no. 1 (2012), 137-151, http://dx.doi.org/10.7494/OpMath.2012.32.1.137

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