I have a multi-variate, continuous function from $R^n$ to $R$, which I can query for its output for any input. I would like to create an interpolation of that function by sampling a subset of the points and create a polynomial representation. I've read literature on univariate function interpolation and it seems using the Chebyshev nodes: http://en.wikipedia.org/wiki/Chebyshev_nodes will do the job, as it converges to the true function as the number of interpolant points increases.

I would like an analogous result for multi-variate functions, where one can choose the samples of the function at the right locations and just interpolate, and the result converges to the function as we increase the number of samples. However I am not able to find any literature on that, is it because it is impossible to extend the result to a multi-variate case? I heard about "chebyshev grids" which is a higher dimensional chebyshev nodes, but does interpolation on the grids retains the convergence property that as the number of samples increase?