[latexpage]
At first, we sample $f(x)$ in the $N$ ($N$ is odd) equidistant points around $x^$:
[
f_k = f(x_k),\: x_k = x^
+kh,\: k=-\frac{N-1}{2},\dots,\frac{N-1}{2}
]
where $h$ is some step.
Then we interpolate points ${(xk,fk)}$ by polynomial
\begin{equation} \label{eq:poly}
P
{N-1}(x)=\sum
{j=0}^{N-1}{a_jx^j}
\end{equation}
Its coefficients ${aj}$ are found as a solution of system of linear equations:
\begin{equation} \label{eq:sys}
\left{ P
{N-1}(x_k) = f_k\right},\quad k=-\frac{N-1}{2},\dots,\frac{N-1}{2}
\end{equation}
Here are references to existing equations: (\ref{eq:poly}), (\ref{eq:sys}).
Here is reference to non-existing equation (\ref{eq:unknown}).