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.