Under the assumption that the structure amplitude of X-ray diffraction from a crystal satisfies the causal Fourier transform condition and appears to be a function with a band-limited spectrum, discrete Hilbert transforms (DHT) linking structure amplitudes having half-integral-valued Miller indices with structure amplitudes having integral-valued indices are obtained. DHT are then used to derive an interpolation formula that permits structure-amplitude reconstruction from samples with half the sampling frequency of the Nyquist rate. Some one-dimensional test calculations are also given.

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Phase extension from lower to higher resolution by using an upgraded TWIN variables algorithm [Hountas & Tsoucaris (1995). Acta Cryst. A51, 754±763] in protein molecules with close to 1000 non-H atoms is presented. Three points of this procedure are of particular interest. (i) The use of a set of auxiliary variables providing a satisfactory ®t for many kinds of constraints: the new algorithm works ef®ciently despite the extreme`dilution' of very limited initial phase information into a much larger set of auxiliary variables. (ii) The extension of this auxiliary variables set beyond the resolution of the observed data, which enhances the phase extension in a so-called`super-resolution' sphere. (iii) The use of the crystallographic symmetry as a new ®gure of merit and as a reliable test for the correctness of the phase-extension process allows an ef®cient screening.