I've stumbled upon a method of extrapolation that I haven't seen before.

We are trying to approximate $f(0)$ for a certain function $f$, which we have only measured at points $x_0, \ldots, x_N$ in an interval $[a,b]$ that does not contain $0$. We have reason to believe that $f$ is analytic in a neighbourhood of a region of $\mathbb C$ (containing both $0$ and $[a,b]$) bounded by a simple positively oriented closed contour $\Gamma$. Suppose we can approximate $1/z$ on $\Gamma$ by a linear combination of $1/(z-x_j)$, say $$ \left| \frac{1}{z} - \sum_{j=0}^N \frac{a_j}{z-x_j} \right| \le \varepsilon \ \text{for}\ z \in \Gamma $$ Then using Cauchy's formula, $$ \left|f(0) - \sum_{j=0}^N a_j f(x_j) \right| \le \frac{M\; \text{length}(\Gamma) \varepsilon}{2\pi}$$

Rather than uniform approximation, it is more convenient to use an $L^2$ approximation. This will let us find the $a_j$ by minimizing a quadratic form. In the case where $\Gamma$ is a circle of radius $r$ centred at $0$, I get a nice closed form: $$ a_j = \frac{r^2 - x_j^2}{r^{2N+2}} \prod_{k \ne j} \frac{x_k (r^2 - x_j x_k)}{x_k - x_j}$$

I can't believe I'm the first to think of this idea. Has anyone seen something like this?