Variation and approximation for Mellin-type operators

Mellin analysis is of extreme importance in approximation theory, also for its wide applications: among them, for example, it is connected with problems of Signal Analysis, such as the Exponential Sampling. Here we study a family of Mellin-type integral operators defined as $$(T_w f)({\tt s})=\int_{\R_+^N} K_w({\tt t}) f({\tt st}){\,d{\tt t} \over \langle{\tt t}\rangle}, \ {\tt s}\in \R_+^N,\ w>0,\eqno \rm{(I)}$$ where $\{K_w\}_{w>0}$ are (essentially) bounded approximate identities, $\langle{\tt t}\rangle:=\prod_{i=1}^N t_i,$ ${\tt t}=(t_1,\dots,t_N)\in \R^N_+$, and $f:\R_+^N\rightarrow \R$ is a function of bounded $\varphi-$variation. We use a new concept of multidimensional $\varphi-$variation inspired by the Tonelli approach, which preserves some of the main properties of the classical variation. For the family of operators (I), besides several estimates and a result of approximation for the $\varphi-$modulus of smoothness, the main convergence result that we obtain proves that $$\lim_{w\to +\infty} V^{\varphi}[\mu(T_w f-f)]=0,$$ for some $\mu>0$, provided that $f$ is $\varphi-$absolutely continuous. Moreover, the problem of the rate of approximation is studied, taking also into consideration the particular case of Fej\'er-type kernels.