A new stable basis for radial basis function interpolation

It is well known that radial basis function interpolants suffer from bad conditioning if the basis of translates is used. In the recent work by Pazouki and Schaback (2011), the authors gave a quite general way to build stable and orthonormal bases for the native space NΦ(Ω) associated to a kernel Φ on a domain Ω⊂Rs. 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 N_Φ(Ω) and in ℓ_{2,w}(X), where X is a set of data sites of the domain Ω. 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 TΦ:N_Φ(Ω)→N_Φ(Ω), 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 TΦ. Finally, using the eigenvalues of this operator, we provide convergence estimates and stability bounds for interpolation and discrete least-squares approximation.