Time-delay polynomial networks and rates 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, and we give an 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.