SUMMARY
The three-level comparator is applied as a tool to formation of vector-indicator of function and its argument and recurrent artificial neuron net (RANN). The modernization of expression in Fourier series due to usage of the vector-indicator components is introduced. The example of RANN for peripheral processing data on the basis of a long-short-term memory is proposed. The dependence of number point shift from order of oldest derivatives in expression is studied. The system equations realizing conditions of minimization a sum of squared deviations from the patterns are written. The processes of transformation on different stage of data acquisition and processing into RANN are considered. Decomposition of function on derivatives and vector-indicator inside RANN is shown. The numerical experiments for analytical learning are done, they demonstrated convergence of analytical learning algorithms independently from first approximation even for oscillating operators.