Given a function $f$ on an interval $(a,b)$ (or alternatively a set of data points ${(x_i,y_i)}$), is there an efficient algorithm to find a rational function of type $(m,n)$ with integer coefficients that (in some way) approximates $f$ (or interpolates between the data points)?
Any references are appreciated.
EDIT
In order to elicit an answer, I would like to reformulate my question and make it more precise.
Assume that $$ f:x\mapsto\frac{\sum_{i=0}^n{a_i x^i}}{1+\sum_{j=1}^m{b_j x^j}} $$ is a rational function of type $(n,m)$ without poles in $[0,1]$ and that $(f_n)_n$ is a sequence of (not necessarily rational) functions that converge uniformly to $f$ on $[0,1]$.
Denote further by $a_i^{(n)}$ and $b_j^{(n)}$ the coefficients that are obtained from sampling the function $f_n$ at $n+m+1$ points $x_\nu\in[0,1]$ and interpolating the points $(x_\nu,f_n(x_\nu))$ with a rational function of type $(n,m)$, i.e. $$ f_n(x_\nu)=\frac{\sum_{i=0}^n{a_i^{(n)} x_\nu^i}}{1+\sum_{j=1}^m{b_j^{(n)} x_\nu^j}},\quad \nu=1,\ldots,n+m+1. $$
I have two questions:
- Can the errors $\left|a_i^{(n)}-a_i\right|$ and $\left|b_j^{(n)}-b_j\right|$ be bounded by some function of the distance between $f_n$ and $f$, e.g. by $\sup_{0\leqslant x\leqslant 1}\left|f(x)-f_n(x)\right|$?
- If the answer to the first question is yes, how does this bound depend on the choice of evaluation sites $x_\nu$? I know that for polynomial interpolation certain choices of evaluation sites enjoy special optimality properties.