  A fast method of an arbitrary high order for approximating volume potentials
is proposed, which is effective also in high dimensional cases. Basis functions
introduced in the theory of approximate approximations are used. Results of
numerical experiments, which show approximation order O(h^8) for the Newton
potential in high dimensions, for example, for n= 200 000, are provided. The
computation time scales linearly in the space dimension. New one-dimensional
integral representations with separable integrands of the potentials of
advection-diffusion and heat equations are obtained.
