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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:

  1. 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 $r(x)$?
  2. 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 $r(x)$?
  3. 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 $\mathbb C^{N+1}$?
  4. Where can I learn more about this topic, since when is this known, ... ?
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  • $\begingroup$ In Barycentric Lagrange Interpolation. Jean-Paul Berrut, Lloyd N. Trefethen., the earliest references seem to be from 1997 of Jean-Paul Berrut. This is a case of an author citing another paper of himself as reference, so "it is known" might only refer to a very small set of people... $\endgroup$ – Thomas Klimpel Feb 21 '15 at 13:59
  • $\begingroup$ It's not clear, in your displayed equation, what $y_j$ denotes. $\endgroup$ – John Hughes Feb 21 '15 at 14:10
  • $\begingroup$ @JohnHughes You mean because $y_j$ is not just a coefficient, but also identical to $r(x_j)$? Maybe you have a good point, and it could be related to my troubles understanding the cited statement. $\endgroup$ – Thomas Klimpel Feb 21 '15 at 14:21
  • $\begingroup$ Actually, I assumed that $y_i$ denoted a variable. I think Trefethen and Berrut use $f_i$ for this, indicating that it's the value that you're trying to interpolate with some function $f$. Now that I think about it, $y_i$ is a perfectly good name too. :) I think if you read the first few pages of the paper cited by Thomas Klimpel, you'll find that the answer to item 1 is "yes". I suspect that "2" is also "yes", and that the proof's identical. Finally, the last page of the cited paper contains a bunch of historical remarks that should address item 4. $\endgroup$ – John Hughes Feb 21 '15 at 17:36
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I found a derivation in one of the papers from 1997 of Jean-Paul Berrut cited in Barycentric Lagrange Interpolation. Jean-Paul Berrut, Lloyd N. Trefethen. The claim is that any rational function $r(x)=\frac{p(x)}{q(x)}$ with $\max(\operatorname{deg}p,\operatorname{deg}q)\leq n$ for $n+1$ distinct interpolation points $x_j$ with $q(x_j)\neq 0$ can be written in barycentric form $$r(x)=\frac{\sum_{j=0}^n\frac{u_j}{x-x_j}r_j}{\sum_{j=0}^n\frac{u_j}{x-x_j}}$$ for $r_j:=r(x_j)$ and suitable weights $u_j$. The derivation is actually quite simple:

Let $l(x)=(x-x_0)(x-x_1)\dots(x-x_n)$ and $w_j=\left.\frac{x-x_j}{l(x)}\right|_{x=x_j}$. The barycentric Lagrange formula represents the denominator as $q(x)=l(x)\sum_{j=0}^n\frac{w_j}{x-x_j}q_j$. For $\tilde{p}(x):=l(x)\sum_{j=0}^n\frac{w_j}{x-x_j}q_jr_j$ we have $$\frac{\tilde{p}(x)}{q(x)}=\frac{\sum_{j=0}^n\frac{w_j}{x-x_j}q_jr_j}{\sum_{j=0}^n\frac{w_j}{x-x_j}q_j}=\frac{\sum_{j=0}^n\frac{u_j}{x-x_j}r_j}{\sum_{j=0}^n\frac{u_j}{x-x_j}}$$ with $u_j:=w_jq_j$, and $\tilde{p}(x)=p(x)$ follows from $\operatorname{deg}p\leq n$ and $q_j=q(x_j)\neq 0$. We have $p(x_j)=q_jr_j$, because $q_j=q(x_j)\neq 0$. So $\tilde{p}(x)$ coincides with $p(x)$ at $n+1$ points, and both are polynomials of degree less than $n+1$, so they are identical.


The article Some New Aspects of Rational Interpolation. Claus Schneider, Wilhelm Werner from 1986 uses the barycentric form for rational interpolation and highlights its advantages, but doesn't seem to waste time or space to state or prove such a basic fact about the barycentric form. It might indeed have been stated and proved for the first time by Jean-Paul Berrut, maybe because people at that time grew tired of basic folklore results which practitioners knew but didn't publish, because the results were not new or non-trivial enough.

A good place to find more information about rational interpolation might be the reference section of this article about rational interpolation, i.e. the articles

  1. Recent developments in barycentric rational interpolation. Jean–Paul Berrut, Richard Baltensperger and Hans D. Mittelmann.
  2. Barycentric rational interpolation with no poles and high rates of approximation. Michael S. Floater, Kai Hormann.
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