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Suppose $x,y \in \mathbb{R}^n$ for some given fixed n.

Consider a kernel $K(x,y) = f(\langle x, y \rangle)$, I'd like to know which functions $f$ admit a finite dimensional feature map. In other words, for $x,y \in \mathbb{R}^n$, what functions $f$ does there exist an $m$ and $\phi: \mathbb{R^n} \rightarrow \mathbb{R}^m$ with

$f(\langle x, y \rangle ) = \langle \phi(x), \phi(y)\rangle?$

I can show that $f$ must be polynomial if $m < 2^n$, but I'm sure there must exist a more comprehensive result.

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