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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?

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  • $\begingroup$ If you accept piecewise polynomial, B-splines will do the job. Numerical interpolation usually displays a local (polynomial) side and a global (piecewise) side. $\endgroup$
    – Denis Serre
    Commented Oct 6, 2014 at 6:23
  • $\begingroup$ 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. $\endgroup$
    – Evan Pu
    Commented Oct 6, 2014 at 14:07

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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].

[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.

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