Rational approximation

Summary

Rational approximation is the approximation of functions by rational functions $p/q$ with prescribed bounds on the degrees of $p$ and $q$. Rational functions can imitate poles, steep fronts, and singular points much better than polynomials of the same number of parameters. A striking case is Newman's theorem: $\abs{x}$ on $[-1, 1]$ has rational approximation error of order $e^{-c\sqrt{n}}$, while polynomials only achieve order $1/n$. Padé approximants and best uniform rational approximations are the main constructions.