  We define the algebra of Colombeau generalized functions on the space of
generalized points of {\mathbb R}^d which naturally contains the tempered
generalized functions. The subalgebra of \mathscr{S}-regular generalized
functions of this algebra is characterized by a pointwise property of the
generalized functions and their Fourier transforms. We also characterize the
equality in the sense of generalized tempered distributions for certain
elements of this algebra (namely those with so-called slow scale support) by
means of a pointwise property of their Fourier transforms. Further, we show
that (contrary to what has been claimed in the literature) for an open subset
\Omega of {\mathbb R}^d, the algebra of pointwise regular generalized functions
\dot{\mathcal G}^\infty(\Omega) equals {\mathcal G}^\infty(\Omega) and give
several characterizations of pointwise {\mathcal G}^\infty-regular generalized
functions.
