Summary: It is well known that radial basis function interpolants suffer from bad conditioning if the basis of translates is used. M. Pazouki and R. Schaback [J. Comput. Appl. Math. 236, No. 4, 575–588 (2011; Zbl 1234.41003)] gave a quite general way to build stable and orthonormal bases for the native space \(\mathcal N_{\Phi}({\Omega})\) associated to a kernel \({\Phi}\) on a domain \({\Omega} \subset \mathbb R^s\). The method is simply based on the factorization of the corresponding kernel matrix. Starting from that setting, we describe a particular basis which turns out to be orthonormal in \(\mathcal N_{\Phi}({\Omega})\) and in \(\ell_{2,w}(X)\), where \(X\) is a set of data sites of the domain \({\Omega}\). The basis arises from a weighted singular value decomposition of the kernel matrix. This basis is also related to a discretization of the compact operator \(T_{\Phi} : N_{\Phi}({\Omega}) \to N_{\Phi}({\Omega})\), \[ T_{\Phi}[f](x) = \int_{\Omega}{\Phi}(x, y)f(y)dy \forall x \in {\Omega}, \] and provides a connection with the continuous basis that arises from an eigendecomposition of \(T_{\Phi}\). Finally, using the eigenvalues of this operator, we provide convergence estimates and stability bounds for interpolation and discrete least-squares approximation.