# How to optimally choose points for multivariable Hermite interpolation?

I have a multi-variate, continuous function $f$ from $R^n$ to $R$, which I can query for its output for any input. I would like to create interpolation polynomial for it.

In one-dimensional case choosing points is relatively straight forward. I have error estimation: $$f(x)-H(x)={\frac {f^{{(K)}}(c)}{K!}}\prod _{{i}}(x-x_{i})^{{k_{i}}},$$ and to choose the interpolation points, I need to solve corresponding minmax problem, as I'm looking to minimize the maximum error for a given number of interpolation points. I'm willing to go with any class of multivariate polynomials. I'm even willing to go with something other that polynomials if helps me interpolate my function, takes into account derivatives and provides theoretical guarantees (for example, in form of error bound).

In multi-variate case I cannot find such error bounds in the literature nor I unable to derive them myself. Any advice on how should I pick the interpolation points for this function?

• Can you not simply do a tensor product for the 1D points to multi-dimensions? It works for Lagrange interpolation. – Vikram Dec 13 '16 at 9:26
• This question might get better answers on Mathoverflow. Also, you should explicitly state that you are looking to minimize the maximum error for a given number of interpolation points. Finally, you should define what class of multivariate polynomials you are interested in, since there is more than one natural possibility. – David Ketcheson Dec 13 '16 at 10:34
• @Vikram I'm note sure a tensor product provides optimal grid, since error bound seems to be different in, say, 2d\3d case if exists at all. Surely, it must exist, but I was unable to find it. – Moonwalker Dec 13 '16 at 17:56
• @DavidKetcheson thank you very much for your useful suggestions! I've updated the question accordingly. – Moonwalker Dec 13 '16 at 17:58
• Ah, I see that you already tried MO: mathoverflow.net/questions/254572/… – David Ketcheson Dec 13 '16 at 18:40