The following question was asked on MSE, but might be more suitable here.

Assume there is a rational function $$ f:x\mapsto \frac{\sum_{i=0}^m{a_ix^i}}{1+\sum_{j=1}^n{b_jx^j}} $$ of type $(m,n)$ with real coefficients and without poles on the interval $[0,1]$. Further assume that there is a sequence $(f_n)_n$ of not-necessarily rational functions that converge to $f$ uniformly on $[0,1]$.

For some choice of evaluation sites $x_k\in[0,1]$ we define real numbers $a_i^{(n)}$, $b_j^{(n)}$ as the coefficients of the rational function that interpolate between the $m+n+1$ points $(x_k,f^n(x_k))$, i.e. $$ f_n(x_k)=\frac{\sum_{i=0}^m{a_i^{(n)}x_k^i}}{1+\sum_{j=1}^n{b^{(n)}_jx_k^j}}. $$

**Question:**

How can the errors $\left|a_i-a_i^{(n)}\right|$ and $\left|b_j-b_j^{(n)}\right|$ be controlled in terms of $\sup_{0\leqslant x\leqslant 1}\left|f(x)-f_n(x)\right|$?

If the coefficients $a_i$, $b_j$ are integers, is there a more efficient way of recovering them from the approximants $f_n$ than computing an approximating rational function with real coefficients and rounding to the nearest integers, or doing a brute force search for the rational function with integer coefficients that minimizes the distance to $f_n$?

In my application, $f$ is the cumulative distribution function of a random variable and $f_n$ are empirical distribution functions based on $n$ simulated realisations of this variable. In a numerical example of type $(4,3)$ I need $\left\|f-f_n\right\|_\infty<10^{-6}$ to obtain interpolating rational functions (based on eight equidistant evaluation sites) with coefficients within $0.5$ of the true ones. By Donsker's theorem this corresponds to about $10^{12}$ realisations which is of course not feasible.

Thanks.

**EDIT:**
As requested by Manfred Weis, I add a few more details about the concrete problem I would like to solve. I have a random variable of which I know *a priori* that its cumulative distribution function (CDF) is rational of type $(m,n)$. My goal is to determine the coefficients of the CDF based on simulations. My strategy is to compute the empirical distribution functions (eCDF's) which are known to converge uniformly to the true CDF, approximate those by rational functions and to use the coefficients of the approximate function as approximations for the coefficients of the true CDF. I am asking for quantitative error bounds for this method or pointers to more efficient procedures.