Mathematical Problems in Engineering
Volume 4 (1998), Issue 1, Pages 59-72
Time-delay polynomial networks and rates of approximation
Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin 78712-1084, TX, USA
Received 5 July 1997; Revised 11 August 1997
Copyright © 1998 Irwin W. Sandberg. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider a large family of finite memory causal time-invariant maps G from an input set S to a set of
-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.