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?



Multivariate polynomial interpolations
The answer to your question highly depends on the domain on which you plan to approximate your function.


For products of compact intervals, *e.g.* rectangles in the plane, some kind of tensor product construction can be used. You should take a look at [1], which describes how bivariate functions are handled in Chebfun [2], a Matlab package for representing and computing with univariate and bivariate functions. There are a few references in the aforementioned paper about convergence results.


For more complicated domains, I would take a look at [3].




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[1] A. Townsend & L. N. Trefethen, An extension of Chebfun to two dimensions, SISC, 35 (2013), pp. C495-C518, <https://people.maths.ox.ac.uk/trefethen/cheb2paper.pdf>.


[2] Chebfun, http:// www.chebfun.org/.


[3] B. N. Ryland & H.Z. Munthe-Kaas, On Multivariate Chebyshev Polynomials and Spectral Approximation on Triangles, <http://www.ii.uib.no/~hans/Chebyshev/Welcome_files/munthekaas09mcp.pdf>.


