International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 99100

The calculation of the inner products from a sampled gradient map D requires even more caution than that of structure factors via electrondensity maps described in Section 1.3.4.4.5, because the functions have transforms which extend even further in reciprocal space than the themselves. Analytically, if the are Gaussians, the are finite sums of multivariate Hermite functions (Section 1.3.2.4.4.2) and hence the same is true of their transforms. The difference map D must therefore be finely sampled and the relation between error and sampling rate may be investigated as in Section 1.3.4.4.5. An examination of the sampling rates commonly used (e.g. one third of the resolution) shows that they are insufficient. Tronrud et al. (1987) propose to relax this requirement by applying an artificial temperature factor to (cf. Section 1.3.4.4.5) and the negative of that temperature factor to D, a procedure of questionable validity because the latter `sharpening' operation is ill defined [the function exp does not define a tempered distribution, so the associativity properties of convolution may be lost]. A more robust procedure would be to compute the scalar product by means of a more sophisticated numerical quadrature formula than a mere grid sum.
References
Tronrud, D. E., Ten Eyck, L. F. & Matthews, B. W. (1987). An efficient generalpurpose leastsquares refinement program for macromolecular structures. Acta Cryst. A43, 489–501.