  Here we provide a unifying treatment of the convergence of a general form of
sampling type operators, given by the so-called Durrmeyer sampling type series.
In particular we provide a pointwise and uniform convergence theorem on
$\mathbb{R}$, and in this context we also furnish a quantitative estimate for
the order of approximation, using the modulus of continuity of the function to
be approximated. Then we obtain a modular convergence theorem in the general
setting of Orlicz spaces $L^\varphi(\mathbb{R})$. From the latter result, the
convergence in $L^p(\mathbb{R})$-space, $L^\alpha\log^\beta L$, and the
exponential spaces follow as particular cases. Finally, applications and
examples with graphical representations are given for several sampling series
with special kernels.
