Is a rational function $f(x)$ of degree $m/n$ fully determined by its values $f(x_j)$ at $m+n+1$ points $x_j$? If yes, how does one go about reconstructing the full $f(x)$ from this discrete data?
If $f$ was instead a polynomial of degree $n$, this is straightforward. For example, by the Discrete Fourier Transformation: $$ f(x) = \sum_{j,k=0}^n \frac{f(\omega^k)\,\omega^{-jk}x^j}{n+1} \quad\text{where}\quad \omega=e^{2\pi i/(n+1)}. $$ Alternatively, given a set of $n+1$ distinct points $x_j$, we may use Lagrange polynomials to recover: $$ f(x) = \sum_{0\leq j\leq n} f(x_j) \prod_{0\leq k\leq n\\\hspace{5pt} k\neq j} \frac{x-x_k}{x_j-x_k}. $$ In both cases, one only needs to evaluate the function at $n+1$ points to get the full polynomial.
Do similar methods exists for rational functions of the general form $$ f(x) = \frac{p(x)}{q(x)} = \frac{p_0 + p_1 x +\dots+ p_m x^m}{1 + q_1x +\dots+ q_n x^n} $$ where one would only need to find $n+m+1$ values of $f$ to fully reconstruct $f$?