A Pade-Chebyshev approximation is a formal transformation of the Chebyshev series. Although the Pade and Pade-Chebyshev approximations contain no more information than the approximations that they were derived from, they are often better. Thus they are a half-way step to full rational approximation. A rational function is a ratio of a polynomial numerator and denominator. A minimax rational approximation minimizes the maximum error over a given interval.
Rational functions appear to be a powerful and underutilized tool in geometry.
Rational approxes are harder to work with, partly since they are not linear and you cannot just linearly combine a set of basis functions. However, they often give much more economical approxes, expecially to functions with kinks, like , or asymptotes, like , than do polynomials. Here is some info on how much better a rational approx is that a polynomial approx for . would seem to be about the best function for polynomially approximating, so this result is interesting. The is because Taylor is suboptimal. The best rational approx, which will have roughly coefficients in both numerator and denominator, has an error of .