Accurate computation of the high dimensional diffraction potential over hyper-rectangles

We propose a fast method for high order approximation of potentials of the Helm-holtz type operator (Delta+kappa^2) over hyper-rectangles in (R^n). By using the basis functions introduced in the theory of approximate approximations, the cubature of a potential is reduced to the quadrature of one-dimensional integrals with separable integrands. Then a separated representation of the density, combined with a suitable quadrature rule, leads to a tensor product representation of the integral operator. Numerical tests show that these formulas are accurate and provide approximations of order (6) up to dimension (100) and (kappa^2=100).