The wikipedia article on Pade approximations says that its the "best" rational approximation of a given function but doesn't elaborate further. I've seen some form of this claim posted around the internet but I can't seem to find anyone that defines whats meant by "best" here.
Pade approximation is often compared to Taylor series expansion and truncation but as I recently learned Taylor series expansion isn't actually ideal. How does Pade approximation compare to Chebyshev approximation? Is Pade approximation actually the best rational aproximation in terms of the uniform norm?
If Pade approximation is not truly the best, what are good techniques for solving the following problem:
Given an "well behaved" (take your pick of regularity constraints here) function $f$ and a range $[a, b]$ find two polynomials $P$ and $Q$ of degree $n$ and $m$ respectively such the the following norm is minimized:
$$\max_{x \in [a, b]}{\lvert f(x) - \frac{P(x)}{Q(x)} \rvert}$$
That feels like the most natural definition of "best" to me personally. What literature exists for this problem?