A Fourier-domain formula for the least-squares projection of a function onto a repetitive basis in N-dimensional space

A theorem concerning the least-squares projection of an arbitrary function onto an infinite basis of translated function is given. The theorem provides an explicit formula for the Fourier transform, of the projected function. The formula has the advantage of being valid for least-squares projections in any N-dimensional space. The expression for the projected function can be approximately inverted, using the discrete Fourier transform, to find the actual basis coefficients