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?