Computing complex polynomial Chebyshev approximants on the unit circle by the real Remez algorithm

In recent decades, several generalizations of the real Remez algorithm to t he complex domain have been proposed. For example, a recent paper by Tseng, [SIAM J. Numer. Anal., 33 (1996), pp. 2017-2049] presents a generalized mu ltiple exchange method for solving Chebyshev approximation problems by poly nomials on the unit circle. His method is particularly efficient when the n umber of extremal points characterizing the optimal solution is close to it s lower bound n + 1. (n - 1 is the degree of the polynomial approximant.) U nder the same assumptions, the aim of this paper is to show that the comple x problem can be solved by considering a real polynomial Chebyshev approxim ation. Hence, we apply the real Remez algorithm and we illustrate the effic iency of this approach by various numerical experiments, e.g., in digital f ilter design.