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

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.


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