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

A study is made of the problem of approximating a nontrivial field by finite polynomials in the free fields. This problem, like the well-known problem of the local saturation of the matrix elements of the commutator of fields, reduces to the solution of a finite system of equations for the $r$-functions (nondiagonal matrix elements of the Heisenberg current). This corresponds to a definite truncation (with respect to the particle number) of the well-known infinite system of axiomatic equations for the $r$-functions of local quantum field theory.