  We elaborate a cut (truncated) Mellin moments (CMM) approach that is
constructed to study deep inelastic scattering in lepton-hadron collisions at
the natural kinematic constraints. We show that generalized CMM obtained by
multiple integrations of the original parton distribution $f(x,\mu^2)$ as well
as ones obtained by multiple differentiations of this $f(x,\mu^2)$ also satisfy
the DGLAP equations with the correspondingly transformed evolution kernel
$P(z)$. Appropriate classes of CMM for the available experimental kinematic
range are suggested and analyzed. Similar relations can be obtained for the
structure functions $F(x)$, being the Mellin convolution $F= C \ast f$, where
$C$ is the coefficient function of the process.
