Let $H$ be a Reproducing Kernel Hilbert Space with elements $f:X\rightarrow \mathbb{C}$, with kernel $K(x, y)$. My question is whether, for some choice of $x_i\in X$, it is the case that $u_i:=K(x_i, \cdot)$ is a basis for $H$. What additional conditions should be imposed on the $x_i$?
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If {z_n} is not a zero set for the Hilbert space (separable) then K(z_n, .) spans, but is not necessarily a Schauder basis. I believe the sequence needs to be an interpolating sequence for the Kernels to form a Shauder basis. 

