Rigorous rationale for the Pade Approximant?

Question

I recently asked a Question for which the only Answer I got was a recommendation to punt and use the Pade Approximant.

This is the first time I recalling seeing this, and intuitively it seems like a good approximation. I have seen and generated several examples where the Pade Approximant of order ${m,n}$ is visually better than the $(m+n)th$ order Maclaurin Series. I've also seen online sources that refer to is as the "best" approximation without explaining in what way. So is the Pade Approximant just a really intuitive heuristic? Or is there some fit metric that it actually optimizes?

Answer 1

There seem to be two questions here:
1) do rational functions give better approximants than polynomials ?
2) does Padé give the best rational approximants ?

Of course, "it depends", but some comments beyond that:

In signal processing, there's a lot of lore, and a 5-foot shelf of books and methods, on FIR and IIR filters -- polynomial and rational approximants in the frequency domain. My non-expert summary is that rational functions can fit "peaky" spactra much better, but can be tricky; don't use degree > 2 in the denominator ... unless you know what you're doing.

On Padé, Numerical Recipes pages 245-247 says

... it relates to analyticity ...
there is, in general, no way to tell how accurate it is.

Answer 2

As far as I have been able to tell, there isn't a rigorous rational for the Padé approximant. The most recent work that I found about these approximants are the two books written by Baker

1. Padé Approximants (Encyclopedia of Mathematics and its Applications)
2. Essentials of Padé Approximants

I haven't found any paper in the archive which answers why Padé approximants are so effective. Most people use Padé approximants inside numerical solvers and observe that the results are better than truncated Taylor series.

A Padé approximant is a particular type of rational approximation whose power series expansion agrees with a given power series to the highest possible order. The $L,M$ Padé approximant is denoted by

$$[L/M] = \frac{P_L(x)}{Q_M(x)}$$

where $P_L(x)$ is a polynomial of degree less than or equal to $L$, and $Q_M(x)$ is a polynomial of degree less than or equal to $M$. The formal power series

$$f(x) = \sum _{j=0}^\infty f_jx^j$$

determines the coefficients by the equation

$$f(x) -\frac{P_L(x)}{Q_M(x)}=O(x^{L+M+1})$$

It is well known that Padé approximations can outperform a truncated Taylor expansion when functions contain poles, because the use of rational functions allows them to be well-represented. However, the following open questions remain

1. The $[L/L+J]$ Padé approximants to any meromorphic function converge in measure within any bounded region of the complex plane as $L$ approaches infinity. Can this result be extended to all entire functions? Are there entire functions which cannot be represented by a Padé approximant?

2. Baker has shown convergence properties for the series of Stieltjes and Polya. Can these results be extended to other infinite series? How far can one extend the convergence properties of Padé approximants?

3. How well can rational functions approximate the Riemann zeta function? Can we form a sequence of Padé approximants that converge uniformly to $\zeta$?

4. Why do Padé approximants work better than other approximants? What is the mathematical reason why Padé approximants generally produce better results than a truncated Taylor series?

There are a lot of open questions about Padé approximants. I am currently in my second year of graduate school (studying applied/computational mathematics) and might be able to investigate some of these.