# Multivariate polynomial interpolations

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?

• If you accept piecewise polynomial, B-splines will do the job. Numerical interpolation usually displays a local (polynomial) side and a global (piecewise) side. – Denis Serre Oct 6 '14 at 6:23
• I do accept them! However it seems that whenever doing anything piece-wise, you end up with a million pieces. I was hoping that for each piece we can have a convergence not by breaking it into more pieces, but by increasing the degree of the polynomial at that piece. At least this way we have a trade-off between #pieces and degree of polynomial. – Evan Pu Oct 6 '14 at 14:07