In various papers, e.g. Random Features for Large-Scale Kernel Machines, Rahimi and Recht introduce the now popular methodology wherein a "low rank" approximation to a stationary, PSD, kernel $K(x,y) = <\phi(X),\phi(y)>$ is constructed by randomly sampling a fourier basis from the spectral density $p$ of the kernel K such that: $K(x,y) \simeq <Z(x),Z(y)>$, where $Z(x) = \frac{2}{\sqrt{d}}[\cos(w_{1}^Tx + b_{1}),\cos(w_{2}^Tx + b_{2}),...,\cos(w_{d}^Tx + b_{d})], w_i \sim p, b \sim U(0,2\pi)$.
I am confused about the potential fact that the approximation to the potentially non periodic feature mapping $\phi$ is now approximated by a vector of periodic features. Thinking of this as a fourier series, I would think someone might specify an interval over which this approximation is valid, though no such discussion appears in the literature so I think I'm missing something. Should I be thinking of this method as approximating the kernel (i.e. the inner product between approximate feature mappings) rather than as well approximating the actual feature mapping itself?