multi-dimensional function from mesured values I am thinking about the posibility of determining continual, differentiable functions
$$f:~\mathbb{R}^n\to\mathbb{R}\quad\textrm{or}\quad g:\mathbb{R^n}\to\mathbb{R}^m$$
by giving a non infinite set of points
$$\left\{(\alpha,f(\alpha))~|~\alpha\in(x_1,\dots,x_n)\right\}\,.$$
Just like it is possible with $m=n=1$.
However i am in a lack of keywords which I can search for. Can you tell me what to search for? And if it is even possible?
 A: I suggest you search for "multi-dimensional interpolation" or "multivariate interpolation".
To save you a bit of time, here is a link to a Wikipedia page on the subject.
Many spline interpolation techniques extend in a natural way to multi-dimensional situations. The concepts are the same, but inventing good notations is a lot more troublesome.
A: In fact, it's impossible even in the simpliest cases. There's vast literature on interpolating functions given finite number of their values, for example, via polynomials.
As a counterexample, you take a set of points $x_k=k$, $f(x_k)=0$, $k=1,\dots,N$. Clearly, a polynomial $\Pi_{k=1}^N(x-k)$ satisfies these conditions. On the other hand, so does the function $\sin (\pi x)$. Even further, a constantly zero function does it, too.
What you speak about (finding functions by some values) works only if you make additional hypothesis on the functions and on the number of point you take. For example, if you say that your function is affine, then you can take $2$ points. One is not sufficient and three or more can be incompatible.
Another thing that physists do, that they take some analytical characteristics. For instance, if I know that my function is a maxwellian with unity dispersion, i.e. $$f(x)=\frac{\rho}{\sqrt{2\pi }}e^{-\frac{(x-x_0)^2}{2 }},$$
then $$\rho = \int_{\Bbb R}f(x)dx,$$
$$\rho x_0 = \int_{\Bbb R}xf(x)dx.$$
So by taking these two integrals I can reconstruct my whole function.
One of keywords you should use is interpolation. You can start here on wiki.
