Structure theorems for nonlinear systems

Abstract

One of the main results is a proposition to the effect that under some typically mild conditions finite sums of the form $$\sum\limits_\ell {K_\ell \sigma } \left[ {\sum\limits_m {\eta _{\ell m} Q_m (\cdot) + \rho _\ell } } \right]$$ are dense in an important sense in the set of shift-invariant approximately-finite-memory mapsG(·) that take a certain type of subsetU ofR intoR, whereR is the set of real-valued functions defined onR n orZ n . Here theQ m (·) are linear, σ is any element of a certain set of nonlinear maps fromR toR, and the κl, ρl, and ηlm are real constants. Approximate representations comprising only affine elements and lattice nonlinearities are also presented.