Abelian and Tauberian theorems for the convolution type transformations

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.