Regresar

On Functions of Bounded (φ, k)-Variation

Abstract:

Given a φ-function φ and k ϵ N we introduce and study the concept of (φ, k)-variation in the sense of Riesz of a real function on a compact interval. We show that a function u :[a, b] → R has a bounded (φ, k)-variation if and only if u(k-1) is absolutely continuous on [a, b]and u(k) belongs to the Orlicz class L φ[a, b]. We also show that the space generated by this class of functions is a Banach space. Our approach simultaneously generalizes the concepts of the Riesz φ-variation, the de la Valleé Poussin second-variation and the Popoviciu kth variation.