Rapidly converging approximations and regularity theory

We consider distributions on a closed compact manifold $M$ as maps on smoothing operators. Thus spaces of certain maps between $\Psi^{-\infty}(M)\to $\mathcal{C}$^{\infty}(M)$ are considered as generalized functions. For any collection of regularizing processes we produce an algebra of generalized functions and a diffeomorphism equivariant embedding of distributions into this algebra. We provide examples invariant under certain group actions. The regularity for such generalized functions is provided in terms of a certain tameness of maps between graded Frech'et spaces. This notion of regularity implies the regularity in Colombeau algebras in the ${\text{maG}}^{\mathrm{\infty }}$ sense.