A New Concept of Multidimensional Variation in the Sense of Riesz and Applications to Integral Operators

In this paper we introduce and study a new multidimensional generalization in the sense of Tonelli of the Riesz–Medvedev φ-variation, proving in particular a multidimensional generalization of the Riesz–Medvedev theorem. We finally discuss an application of such concept of variation to some approximation problems: in particular, we obtain some estimates and convergence results by means of convolution integral operators in the space of functions of bounded φ-variation.