A Wiener-type Tauberian theorem for generalized functions of slow growth

The paper is devoted to the extension of Wiener-type Tauberian theorems to the case of generalized functions of slow growth. A functional is shown to have asymptotics (in the weak sense) if and only if it has asymptotics on a 'test' function whose Mellin transform is bounded alway from zero in a certain strip of the complex plane related to the order of the functional in question. Applications of this result are also considered; in particular, several theorems on the lack of compensation of the singularities: of holomorphic functions are proved.