Best causal mathematical models for a nonlinear system

We provide new causal mathematical models of a nonlinear system S which are specifications of a nonlinear operator P_p of degree p=1,2,.... The operator P_p is determined from a special orthogonalization procedure and minimization of the mean squared difference between outputs of S and P_p. As a result, these models have smallest possible associated errors in the class of such operators P_p. The causality condition is implemented through the use of specific matrices called lower trapezoidal. The associated computational work is reduced by the use of the orthogonalization procedure. We provide a strict justification of the proposed approach including theorems on an explicit representatoin of the models´ parameters, and theorems on the associated error representation. The possible extensions of the proposed approach and its potential applications are outlined.