On the first kind Volterra convolution equation

We will consider the following integral equation on the positive half-line:

$\int \limits_0^x V(x-t) f(t) dt = g(x), \qquad x>0,$

It is supposed that the kernel $V$ is non-increasing and absolutely continuous, $V(0) > 0$, $V(+ \infty) =0$, $g$ is absolutely continuous and $g(0) = 0$․

This equation is considered as a non-correct problem. It has a theoretical as well as an applied interest in the signal processing theory and etc. The equation can be reduced to the recovery equation, well known in the probability theory. With an application of previous results of the author and his coauthors, it is described the structure of a solution and its asymptotic property. Also, it will be provided a simple way for the number-theoretic solvability of the equation.