An Approximation Error Lower Bound for Integer Polynomial Minimax Approximation

The need to solve a polynomial minimax approximation problem appears often in science. It is especially common in signal processing and in particular in filter design. The results presented in this paper originated from a study of the finite wordlength restriction in the FIR digital filter design problem. They are, however, much more general and can be applied to any polynomial minimax approximation problem in which the polynomial coefficients are constrained to a finite set of numbers. The finite set restriction introduces a nonzero lower bound to the approximation error. For any given non-trivial function that is to be approximated there is a nonzero lower bound below which it is not possible to go, no matter how large the polynomial degree n. For practical purposes it is very useful to know this lower bound because it can be used to substantially increase the speed of the branch-and-bound algorithm that gives the optimal integer coefficients. A method for computing such a bound is presented in the paper.