  In this paper we give a unitary approach for the simultaneous study of the
convergence of discrete and integral operators described by means of a family
of linear continuous functionals acting on functions defined on locally compact
Hausdorff topological groups. The general family of operators introduced and
studied includes very well-known operators in the literature. We give results
of uniform convergence, and modular convergence in the general setting of
Orlicz spaces.The latter result allow us to cover many other settings as the
$L^p$-spaces, the interpolation spaces, the exponential spaces and many others.
