$C_{1}>0\quad C_{2}>0.$ Hence the function $\nu(y)$ has the form $$\nu(y)=\frac{1}{\alpha}|y|^{-\alpha}(C_{1}1_{y<0}-C_{2}1_{y>0}).$$Let us introduce the functions $$k_{0}(y)=\frac{1}{\alpha(\alpha-1)}|y|^{1-\alpha}(C_{1}1_{y<0}+C_{2}1_{y>0}),$$ where $0<\alpha<2,\quad \alpha{\ne}1.$ When $\alpha
=1$ we have $$k_{0}(y)=
-\mathrm{log}|y|\,(C_{1}1_{y<0}+C_{2}1_{y>0}).$$ It means that the conditions of Theorem 2.1 are fulfilled. Hence the generator $L$ for the stable processes admits the convolution type representation (2.1),(2.2).\
**Proposition 2.3.** *The kernel $k(y)$ of the operator $S$ in representation $(2.1)$ for the stable processes has form $(2.31)$, when $1{\leq}\alpha<2$, and has form $(2.35)$ when $0<\alpha<1$.*\
**Example 2.2.** The variance damped Levy processes .\
For the variance damped Levy processes we have $A=0,\quad
\gamma=\overline{\gamma}$ and $$\nu^{\prime}(y)=C_{1}e^{-\lambda_{1}
|y|}|y|^{-\alpha-1}1_{y<0}+C_{2}e^{-\lambda_{2}|y|}y^{-\alpha-1}1_{y>0},$$where $C_{1}>0\quad C_{2}>0,\quad \lambda_{1}>0,\quad \lambda_{2}>0 \quad
0<\alpha<2.$ It follows from (2.43) that the conditions of Proposition 2.1 are fulfilled when $\alpha{\ne}1.$ If $\alpha=1$ the conditions of Proposition 2.2 are fulfilled. Hence the generator $L$ for the variance damped Levy processes admits the convolution type representation (2.1),(2.2) and the kernel $k(y)$ is defined by formulas (2.30),(2.31), when $1{\leq}\alpha<2$,and by formula (2.35) when $0<\alpha<1$.\
**Example 2.3.** The variance Gamma process .\
For the variance Gamma process we have $A=0,\quad
\gamma=\overline{\gamma}$ and $$\nu^{\prime}(y)=C_{1}e^{-G
|y|}|y|^{-1}1_{y<0}+C_{2}e^{-M|y|}y^{-1}1_{y>0},$$where $C_{1}>0\quad C_{2}>0,\quad G>0,\quad
M>0.$ It follows from (2.44) that the conditions of Proposition 2.2 are fulfilled and the generator $L$ of variance Gamma 