  We study the localization properties of generalized functions defined on a
broad class of spaces of entire analytic test functions. This class, which
includes all Gelfand--Shilov spaces $S^\beta_\alpha(\R^k)$ with $\beta<1$,
provides a convenient language for describing quantum fields with a highly
singular infrared behavior. We show that the carrier cone notion, which
replaces the support notion, can be correctly defined for the considered
analytic functionals. In particular, we prove that each functional has a
uniquely determined minimal carrier cone.
