Latex and the Dollar Character

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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 $\{(x_k,f_k)\}$ by polynomial
\begin{equation} \label{eq:poly}
P_{N-1}(x)=\sum_{j=0}^{N-1}{a_jx^j}
\end{equation}
Its coefficients $\{a_j\}$ 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}).

Renders as:

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 \{(x_k,f_k)\} by polynomial

(1)   \begin{equation*} P_{N-1}(x)=\sum_{j=0}^{N-1}{a_jx^j} \end{equation*}

Its coefficients \{a_j\} are found as a solution of system of linear equations:

(2)   \begin{equation*} \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: (1), (2).
Here is reference to non-existing equation (??).

\begin{tikzpicture}
[+preamble] \usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
[/preamble] \begin{axis}
\addplot3[surf,domain=0:360,samples=40] {cos(x)*cos(y)};
\end{axis}
\end{tikzpicture}

Renders as:

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