Convolution equations in certain Banach spaces

For a Banach space $E$ and $p > 0$, the following problem is considered: how to identify a finite Borel measure $\mu$ on $E$ by means of the potential $g(a) = \int_E {\vert\vert x - a\vert{\vert^p}d\mu (x),a \in E}$. The solution for infinite-dimensional Hilbert spaces is based on limit correlations between the Fourier transforms of finite-dimensional restrictions of $g$ and $\vert\vert x\vert{\vert^p}$. For finite-dimensional subspaces of ${L_p}$, the Levy representation of norms is used.