This article about polynomial interpolation claims that (it is known that) every rational function may be represented in barycentric form: $$r(x)=\frac{\sum_{j=0}^N\frac{w_j}{x-x_j}y_j}{\sum_{j=0}^N\frac{w_j}{x-x_j}}$$ What about the rational function $r(x,y)=\frac{xy}{x^2+y^2}$? But even if I accept that we are talking only about univariate functions, I would still like to know a bit more about the cited statement:
- Is this statement strictly true for $\mathbb R$, i.e. for any rational function $r(x)=\frac{p(x)}{q(x)}$ with $p,q\in\mathbb R[x]$ (polynomials with real coefficients) there are $w_j, x_j, y_j \in\mathbb R$ representing $r(x)$?
- Is this statement strictly true for $\mathbb C$, i.e. for any rational function $r(x)=\frac{p(x)}{q(x)}$ with $p,q\in\mathbb C[x]$ (polynomials with complex coefficients) there are $w_j, x_j, y_j \in\mathbb C$ representing $r(x)$?
- For $r(x)=\frac{p(x)}{q(x)}$ and $N=\operatorname{deg}p+\operatorname{deg}q$, is the set of $x_j\in\mathbb C$ for which appropriate $w_j,y_j\in\mathbb C$ exist open and dense in $\mathbb C^{N+1}$?
- Where can I learn more about this topic, since when is this known, ... ?