Pointwise characterizations in generalized function algebras

Abstract

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.