Approximation in variation by homothetic operators in multidimensional setting

The aim of the present paper is to obtain convergence results and rates of approximation for a class of convolution integral operators of homothetic type in the space of functions of bounded phi-variation. In particular we work with the phi-variation in the sense of Musielak-Orlicz in multidimensional setting introduced in [L. Angeloni-G.Vinti, "Convergence and rate of approximation for linear integral operators in BV^phi-spaces in multidimensional setting", J. Math. Anal. Appl., 349 (2009), 317--334].The rate of approximation is also investigated, by means of suitable Lipschitz classes.