Kernel approximation of a function known only point-wise?

Assume that I have a set of $N$ points $x_i, i=1,…,N,$ in some space $mathbb{R}^D$, and corresponding point-wise (scalar) function evaluations $f(x_i)$. It is my goal to approximate the unknown function $f(x)$ with RBF kernels:

$$tilde{f}(x)=sum_{i=1}^N w_i k(x,x_i)$$

where $k(x,x_i)$ is a RBF kernel centred on $x_i$. It may seem intuitive to set $w_i=f(x_i)$, but then I will usually not reclaim $f(x_i)$ due to the influence of other basis functions. Towards this end, I have a number of questions, and would appreciate it if you could answer some of them:

1. Is there a procedure you can recommend for finding the weights $w_i$ so that $tilde{f}(x_i)=f(x_i)$?
2. Can this procedure work with higher-dimensional output (i.e., $f:mathbb{R}^D rightarrow mathbb{R}^E,E>1$)?
3. Are there any references or lectures you can recommend on the topic?