On functions of bounded (ϕ,k)-variation

Abstract

Given a ϕ-function ϕ and k∈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 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 Vallée Poussin second-variation and the Popoviciu k-th variation.