Fourier transformation by smooth interpolation

A method is presented for efficiently computing the Fourier transform of a function, such as (charge density)1/3, known numerically at a number of points in the unit cell of a crystal. The method utilizes an implicit interpolant which would be maximally smooth, consistent with the data. However, the interpolant is not computed, but the Fourier coefficients are obtained directly. Specialization of the formulas to a uniform grid is made and results in an enormous saving of computation.