A Wiener-type Tauberian theorem for generalized functions of slow growth. (English. Russian original) Zbl 0953.40004

Drozhzhinov, Yu. N.; Zav’yalov, B. I.

Sb. Math. 189, No. 7, 1047-1086 (1997); translation from Mat. Sb. 189, No. 7, 91-130 (1998).

Authors’ abstract: 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 away 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.

Reviewer: T.D.Todorov (Trieste)

MSC:

40E05	Tauberian theorems
41A60	Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
46E15	Banach spaces of continuous, differentiable or analytic functions

Keywords:

tempered distribution; asymptotic expansion; Banach space; Tauberian theorems; Mellin transform