  In this paper we study convergence results and rate of approximation for a
family of linear integral operators of Mellin type in the frame of
$BV^{\varphi}(\mathbb{R}^N_+)$. Here $BV^{\varphi}(\mathbb{R}^N_+)$ denotes the
space of functions with bounded $\varphi-$variation on $\mathbb{R}^N_+$,
defined by means of a concept of multidimensional $\varphi-$variation in the
sense of Tonelli.
