Asymptotically homogeneous generalized functions and boundary properties of functions holomorphic in tubular cones

We introduce and study a spherical representation of generalized functions and use it to give a complete description of asymptotically homogeneous generalized functions in the case when the order is non-critical and sufficient conditions when it is critical. Generalized functions of slow growth (tempered distributions) that have (quasi-)asymptotics at infinity in the asymptotic scale of regularly varying functions are said to be asymptotically homogeneous. In particular, all homogeneous generalized functions are asymptotically homogeneous. We apply our results to the study of singularities of holomorphic functions in tubular domains over cones.