Convergence and rate of approximation in $B{V}^{\phi }({\mathbb{R}}_{+}^N)$ for a class of Mellin integral operators

Abstract

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 $B{V}^{\phi }({\mathbb{R}}_{+}^N)$. Here $B{V}^{\phi }({\mathbb{R}}_{+}^N)$ denotes the space of functions with bounded $\phi -$variation on ${\mathbb{R}}_{+}^N$, defined by means of a concept of multidimensional $\phi -$variation in the sense of Tonelli.