Corresponding RKHS of Common Kernels A kernel, $k(x_1, x_2)$, has the interesting property that it may be represented as the dot product in a reproducing kernel hilbert space (RKHS), $\phi(x_0)\phi(x_1)$. I know that for the gaussian kernel $\phi$ is infinite dimensional and other properties of kernels but do not have an explicit representation for $\phi$.
I wish to know the explicit representation for $\phi$ of common kernels eg gaussian, periodic, matern etc. 
(Links to papers or books would be great!)
 A: Interesting. Why would you want to know that?
At least the "lifting" function of the polynomial kernel is well known (and on wikipedia): https://en.wikipedia.org/wiki/Polynomial_kernel
Two very good books on the topic:


*

*Kernel Methods for Pattern Analysis, John Shawe-Taylor, Nello Cristianini

*An Introduction to Support Vector Machines and Other Kernel-based Learning Methods,  Nello Cristianini, John Shawe-Taylor


Since they are both from the same authors, they correlate somewhat. I would start with the first one if you are more into kernels than SVMs with kernels. It also has kernel expansions for other types than the polynomial kernel. 
Btw: When used, the rbf kernel is normally not infinite-dimensional. The regulariation of the underlying modelling problem usually imposes finite dimensionality. Unfortunately this is how far it was explained to me. :/
A: For anyone else who was interested in this, there is a paper by Xu et al (2006) that is good. I hope it might help someone in the future. 
In case the DOI link above breaks, the paper is called 
J. Xu et al, "An explicit construction of A reproducing gaussian kernel hilbert space," in 2006, . DOI: 10.1109/ICASSP.2006.1661340.
