The random Fourier feature approach that @betanalpha mentioned above seems to work nicely in Stan–here’s a little example I put together which should also probably be turned into a case study at some point: https://bitbucket.org/flaxter/random-fourier-features-in-stan
Regarding the discussion of inducing point type approaches, it’s worth mentioning also that in practice, good ol’ RBF networks can work pretty well, see a recent publication where we used them in Stan: https://arxiv.org/abs/1705.04293
…which is not to say that the explicit expansion you’re mentioning isn’t neat! I was very interested to discover it for a recent paper I’m revising at the moment (https://arxiv.org/abs/1610.08623) where we needed an explicit Mercer expansion. Fasshauer has written a whole book with McCourt about these types of things, and it’s worth checking it out: http://www.worldscientific.com/worldscibooks/10.1142/9335