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 locali zed in a small region of the real space, if its Fourier transform is likewi se localized in reciprocal space, and if it is a periodical function which possesses a symmetry. The fixed-scale analysis consists in the decompositio n of a studied distribution into a sum of copies of the same standard signa l, but shifted into nodes of a grid in the unit cell. For a specified stand ard 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 dec omposition are calculated; what is the best fixed-scale approximation in th e case that the exact decomposition does not exist. The interrelations betw een the fixed-scale decomposition and the phase problem, automatic map inte rpretation and density-modification methods are pointed out.