On Bi-Dimensional Second µ-Variation

Jurancy Ereú 1 , José Giménez 2  and Nelson Merentes 3
  • 1 DEPARTAMENTO DE MATEMÁTICAS DECANATO DE CIENCIAS Y TECNOLOGÍA UNIVERSIDAD CENTROCCIDENTAL LISANDRO ALVARADO BARQUISIMETO, VENEZUELA
  • 2 DEPARTAMENTO DE MATEMÁTICAS FACULTAD DE CIENCIAS UNIVERSIDAD DE LOS ANDES MÉRIDA, VENEZUELA
  • 3 ESCUELA DE MATEMÁTICAS UNIVERSIDAD CENTRAL DE VENEZUELA CARACAS, VENEZUELA

Abstract

In this paper, we present a generalization of the notion of bounded slope variation for functions defined on a rectangle Iba in ℝ2. Given a strictly increasing function µ-defined in a closed real interval, we introduce the class BVµ,2 (Iba ), of functions of bounded second µ-variation on Iba ; and show that this class can be equipped with a norm with respect to which it is a Banach space. We also deal with the important case of factorizable functions in BVµ,2 (Iba ) and finally we exhibit a relation between this class and the one of double Riemann-Stieltjes integrals of functions of bi-dimensional bounded variation.

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