Geodesic
Abelian and Tauberian theorems for the convolution type transformations
Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 1 (1992), pp. 3-14 
A.-R. Isam. Abelian and Tauberian theorems for the convolution type transformations. Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 1 (1992), pp. 3-14. http://geodesic.mathdoc.fr/item/UZERU_1992_1_a0/
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In this paper we have receved analogies of Abelian and Tauberian theorems for the generalization Laplace transformations, namely the following transformation: $$f(s)=\int\limits^{\infty}_0 \omega(st, \gamma)d\alpha(t),$$ where the sequence is constructed $\gamma=\{\gamma_u\},$ $$\gamma_0=0\leq\gamma_1\leq\gamma_2\leq\ldots \leq\ldots,~\sum{1/ \gamma_u }=\sum{1/ \gamma_u^2}\leq\infty,$$ the function $\omega(t, \gamma)$ summarized the nucleus of Laplace transformation.

[1] D. V. Widder, The Laplase transform., 2 ed., Prinston, 1946, 412 pp. | MR

[2] G. V. Badalyan, “Primenenie preobrazovaniya tipa svertki k teorii obobschennoi problemy momentov Stiltesa”, Izv. AN SSSR. Ser. matem., 31 (1967), 491–530 | Zbl

[3] G. V. Badalyan, Zh. A. Badalyan, “Ob obschei skhodimosti odnogo preobrazovaniya tipa svertki”, Mezhvuz. sb. Armenii, Matematika, 1991, no. 7 | Zbl