Finite-particle approximations of a local field and the local saturation problem

Problem of approximating the non-trivial field by finite polynomials over free fields is considered. This problem as well as the well-known problem of local saturation of matrix elements of the commutator of fields is reduced to solving the infinite system of equations for the r-functions (non diagonal matrix elements of the Heisenberg current). The latter corresponds to a definite way (with respect for the number of particles) of breaking off the well-known infinite system of axiomatic equations for the r-functions in the local quantum field theory