Essentially nonlinear interaction Lagrangians and nonlocalized quantum field theory

A formal scheme is proposed for considering the quantum theory of a single-component scalar field $\varphi(x)$ with essentially nonlinear Lagrangian interaction of the form
$$ L_I(x)=g\sum_{n=0}^\infty\frac{u_n}{n!}\,{:}\varphi^n(x){:}, $$
where $u_n$ is a sequence of numbers which satisfy certain general equations. Reasons are given in favor of the fact that within the bounds of the proposed scheme, a form factor can be selected such that the $S$-matrix constructed for Lagrangian interaction $L_I(x)$ should be finite and unitary in every order of perturbation theory.