Time-delay polynomial networks and quality of approximation

We consider a large family of finite memory causal time-invariant maps G from an input set S to a set of R-valued functions, with the members of both sets of functions defined on the nonnegative integers. We markedly improve recent results by giving an explicit upper bound on the error in approximating a G using a two-stage structure consisting of a tapped delay line and a static polynomial network N. This upper bound depends on the degree of the multivariable polynomial that characterizes N. Also given is a lower bound on the worst case error in approximating a G using polynomials of a fixed maximum degree. These upper and lower bounds differ only by a multiplicative constant. We also give a corresponding result for the approximation of not-necessarily-causal input-output maps with inputs and outputs that may depend on more than one variable. This result is of interest, for example, in connection with image processing