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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?

FirefoxMetzger
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  • It isn't possible with $m = n = 1$ without further restrictions. – Michael Albanese Aug 14 '13 at 10:08
  • I don't know. On the one hand I would agree as there are many functions that go trough $(0,0)$ and $(1,1)$. However physicists came up with lots of equations from R^n->R (preshure P=F/A for example) just by having mesured values and I wonder whether there are specific rules behind finding these functions. – FirefoxMetzger Aug 14 '13 at 10:17
  • Well, there are many functions $f$ that satisfy $f(0) = 0$ and $f(1) = 1$, in fact there are more such functions than there are real numbers, but that's beside the point. When you refer to physical laws, there are automatic restrictions such as continuity, differentiability, homogeneous of a certain degree, etc. If you want to work with the same restrictions, you should explicitly say so in your question. – Michael Albanese Aug 14 '13 at 10:24
  • Yes, you are right. I foregot to mention that and will fix it. Besides I still have no idea what to ask google to get more Information about this kind of problem. Any advice? – FirefoxMetzger Aug 14 '13 at 10:36
  • As others have pointed out, the given values do not uniquely determine a function. I assume that you just want to find some function that interpolates the given values. If my assumption is correct, see my answer below. – bubba Aug 14 '13 at 10:47

2 Answers2

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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.

bubba
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  • It seems like "multi-dimensional interpolation" was the magic keyword I was looking for which finaly gives some good feedback from google. The only think I am still unsure about is how you get from a set of points to "err... this looks like a linear function" in a correct mathmatic way. But I guess this is another question. – FirefoxMetzger Aug 14 '13 at 10:56
  • You can do least squares fitting with a linear function. Then, if the fitted linear function doesn't deviate too far from the given data points, you could conclude that the function is (more-or-less) linear. – bubba Aug 14 '13 at 11:03
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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.

TZakrevskiy
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