Application

The quintessence of the importance of can be illustrated by the fact that can be used for generating the best minimax approximation of a unit square pulse [1, 4]:

Chebyshev polynomials, for the same order, give a far inferior approximation (thin line) of the square pulse as shown in the following figure.

Elliptic rational functions contain the one free parameter that is used to adjust the slope of the pulse approximation. Chebyshev polynomials have no such parameter.

The pulse approximation with elliptic rational functions has many important applications in analog and digital signal processing and system design. Symbolic computation of elliptic rational functions and powerful symbolic algebra environments such as Mathematica, made many successful industrial designs possible [4, 5, 6, 7, 8]. These designs produced robust systems (such as analog and digital filters), shortened time to market, and helped designers make cost-effective solutions.