This paper is an extension of earlier papers [8, 9] on the "native" Hilbert spaces of functions on some domain Ω(∪)Rd in which conditionally positive definite kernels are reproducing kernels. Here, the focus is on subspaces of native spaces which are induced via subsets of Ω, and we shall derive a recursive subspace structure of these, leading to recur-sively defined reproducing kernels. As an application, we get a recursive Neville-Aitken-type interpolation process and a recursively defined orthogonal basis for interpolation by translates of kernels.