# Multivariable polynomial decomposition

There is a beautiful theorem that allows us to decompose a polynomial of several variables into the sum of polynomials of a single variable.

Let $P_{N}(x)$ be a multivarible polynomial of degree $N$, where $x = (x_1,...,x_n) \in \mathbb{R}^n$. Let $\Omega$ be a finite subset of the unit sphere $S^{n-1} = \{ \omega \in \mathbb{R}^n \mid \omega \cdot \omega = 1 \}$ such that $$| \Omega | \geq { N + n -1 \choose n-1 }$$ and points in $\Omega$ are in general position. Then there exist polynomials $P_{N,\omega}$ of a single variable of degree $N$ such that $$P_N(x) = \sum\limits_{\omega \in \Omega} P_{N,\omega} (\omega \cdot x)$$

In other words this theorem allows us to represent a polynomial of a vector variable $x$ as a sum of polynomials of its projections $\omega \cdot x$ on a finite number of directions. My question is how to proove this theorem?

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Do you recall where you came across this theorem, or what it's called? –  anon Mar 13 '12 at 11:07
@anon, this property of a set $\Omega$ is called $N$-solvability. I came across it in one paper on multidimensional moment's problem. I found some references on T. Natterer's book "The mathematics of computerized tomography". In this book the similar property is called $N$-resolvability and it is related with spherical harmonics. –  Nimza Mar 13 '12 at 11:22
1. Can you give an example? 2.Can this be extended to $x\in \mathbb{C}^n$? 3. Would it further simplify the result, if $P_N(x)$ would be homogenous? @anon, can you help? I would be glad to read your opinion... –  draks ... Jul 19 '12 at 21:07
I have a related problem. Did you make any progress on this question? –  tgoossens Nov 1 '14 at 10:02
@tgoossens I think that unfortunately the answer is negative –  Nimza Nov 1 '14 at 23:32