# reference request – Kernels with finite dimensional feature spaces

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.