Basis function of a given kernel

Question

We know that any kernel $\mathbf{K}(\mathbf{x},\mathbf{x}')$ can be written as the inner product of the basis functions $\langle\phi(\mathbf{x}), \phi(\mathbf{x}')\rangle$. Is there a way to derive $\phi(\mathbf{x})$ for a particular kernel, say the Matérn kernel: $${\displaystyle C_{\nu }(d)=\sigma ^{2}{\frac {2^{1-\nu }}{\Gamma (\nu )}}{\Bigg (}{\sqrt {2\nu }}{\frac {d}{\rho }}{\Bigg )}^{\nu }K_{\nu }{\Bigg (}{\sqrt {2\nu }}{\frac {d}{\rho }}{\Bigg )}}$$? ($d = ||\mathbf{x} - \mathbf{x}'||$ is the Euclidean distance between $\mathbf{x}$ and $\mathbf{x}'$)

Answer 1

This question has already be answered. I believe your question is, given any kernel, $K(x, x')$, can you always derive the feature map, $\phi(x)$?. If so, here is the answer:

https://stats.stackexchange.com/questions/436971/is-it-always-possible-to-find-the-feature-map-from-a-given-kernel