Say, I have sampled a complex multi-dimensional function several times. I would like to build a continuous model of this function - how do I express it? The purpose is to build a mathematical model of an unknown empirical function. I know I could simply use some function fitting method, but I need some general approach that offers maybe not the best, but a close and smooth approximation. It would be great if this approach depended on the number of sampled points: the more sampled points there are, the better the approximation. Linear approximation is unsuitable, I need at least quadratic smoothness.
The points are sampled in random manner, not on a lattice. For lattices I could use sinc-function interpolation, but lattice sampling is not possible in my case.
There are numerous methods to do that: you can try to create a Vandermonde matrix, solve it and obtain the polynomial, the degree of which will depend on the number of points. You can also obtain the same result using Lagrange polynomial interpolation or Newton's divided differences. A method of undetermined coefficients will work as well.