Processing of small-angle scattering data with linear prolate spheroidal functions

A system of linear prolate spheroidal functions is used for the parameterization of processing of small-angle scattering (SAS) data in the application for monodisperse systems. Such systems of functions possess a unique property--double orthogonality. It is shown that, in terms of the amount of information extracted from the experimental data, the proposed method is optimal among all other methods based on the expansion of the scattering intensity function in a complete system of orthogonal functions. It is found that the number of series coefficients determined directly from the experimental data coincides with the number of independent parameters of the problem, according to the Kotel'nikov information theorem. The higher-order coefficients, which allow the extrapolation of the scattering curve beyond the measurement interval, can be determined by applying regularization algorithms. Unlike the methods developed previously, this regularization does not affect the information obtained from the experimental data. The method proposed is illustrated with model examples. The algorithms described, including the compact procedure of calculating the spheroidal functions, are used in the computer programs