On the Representation of Infinitely Divisible Distributions on Locally Compact Abelian Groups

Let $X$ be a locally compact abelian separable metric group and $Y$ the group of characters on $X$ be a locally compact abelian separable metric group and $Y$. For any $x\in X$, $y\in Y$ let $(x,y)$ be the value of the character $y$ at $x$. It is shown that the characteristic function $\tilde\mu$ of any infinitely divisible distribution $\mu$ on $X$ has the form
$$ \tilde\mu(y)=\left( {x_0,y}\right)\tilde\lambda (y)\exp\left\{{\int {[(x,y)-1-ig(x,y)]dF(x)-\Phi(y)}}\right\}, $$
where $x_0$ is an element of $X$, $\tilde\lambda$ is the characteristic function of the normalised Haar measure $\lambda$ of a compact subgroup, $g$ is a special function on $X\times Y$ not depending on $\mu $, $F$ is a measure with finite mass outside every neighbourhood of the identity of $X$ which integrates $1-\operatorname{Re}(x,y)$ for each $y\in Y$, and $\Phi$ is a non-negative continuous function on $Y$ satisfying the identity
$$ \Phi \left( {y_1 + y_2 } \right) + \Phi \left( {y_1 - y_2 } \right) = 2\left[ {\Phi \left( {y_1 } \right) + \Phi \left( {y_2 } \right)} \right],\quad y_1 ,y_2 \in Y. $$
This is an extension of an earlier result of K. R. Parthasarathy et al. [1].