# Faber polynomials versus other polynomial approximations

Faber polynomials appear to be used to algorithmically approximate the complex exponential function $\exp(-ix)$ for example as in the case of the time evolution in quantum mechanics, for a Hermitian matrix $H$ and time $t$, an approximation to $\exp(-iHt)$ . In approximation theory, however it is very common to use, e.g., the Chebyshev polynomials in combination with a Taylor series approach to first take the Taylor series approximation to the exponential function, second truncate it, and then third approximate the monomials with a Chebyshev polynomial. Since I am very unfamiliar with the Faber polynomial approach, I was wondering whether there is a benefit using it and whether it gives better bounds for approximating the complex exponential?