Fixed-scale wavelet-type approximation of periodic density distributions

For a chosen unit cell, a function defined in real space (a standard signal) is considered as a crystallographic wavelet‐type function if it is localized in a small region of the real space, if its Fourier transform is likewise localized in reciprocal space, and if it is a periodical function which possesses a symmetry. The fixed‐scale analysis consists in the decomposition of a studied distribution into a sum of copies of the same standard signal, but shifted into nodes of a grid in the unit cell. For a specified standard signal and grid of the permitted shifts in the unit cell, the following questions are discussed: whether an arbitrary function may be represented as the sum of the shifted standard signals; how the coefficients in the decomposition are calculated; what is the best fixed‐scale approximation in the case that the exact decomposition does not exist. The interrelations between the fixed‐scale decomposition and the phase problem, automatic map interpretation and density‐modification methods are pointed out.