Squared and absolute errors in optimal approximation of nonlinear systems. (Q1421411)

Consider a nonlinear, discrete-time system \(y=Gu\) and a model \(\hat{y}=Mu\). The problem is to find \(M\) solving \(\inf_M\lim_{n\to\infty} n^{-1}\sum_{t=0}^{n-1} | Gu(t)-Mu(t)| ^p\) where \(p=1\) or \(p=2\) and where \(M\) ranges over the set of models which can be either polynomial, \(Mx=\sum_{i\geq0}b(i)x^i\), or linear time invariant, \(Mu(t)=\sum_{i\in{\mathbb Z}} f(i)u(t-i)\). Note that the problems are linear in the model parameters \(b(i)\) or \(f(i)\). By imposing some extra (smoothness) conditions on the model classes and the given system, existence and uniqueness results are obtained for these problems. In the case of a polynomial model, a link with classical \(L_2\) and \(L_1\) approximation-theoretic results in the interval \([-1,1]\) are included.