I know how to approximate a real function of a single real variable using Bernstein polynomials. But I do not understand how to extend the procedure to functions of two variables.
I am guessing that the equations in the following image can be used. Am I right? Please correct me if I am wrong.
Your equation contains some typographical errors, which is why I prefer to write out the reasoning in full.
On the unit square you can reason as follows \begin{equation} f(x,y) \approx \sum_{i=0}^m \binom{m}{i} x^i (1-x)^{m-i} f(x_i,y), \end{equation} where $x_i = \frac{i}{m}$, and each $f(x_i,y)$ should be approximated with its own Bernstein polynomial in $y$. You have \begin{equation} f(x_i,y) \approx \sum_{j=0}^n \binom{n}{j} y^j (1-y)^{n-j} f(x_i,y_j), \end{equation} where $y_j = \frac{j}{n}$. If you put these expressions together, you have a certain way of getting the right result and a good way of generalizing to higher dimensions.