A new stable basis for radial basis function interpolation

It is well-known that radial basis function interpolants suffer of bad conditioning if the basis of translates is used. In the recent work [12], 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 NΦ(Ω) and in `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 : NΦ(Ω)→ NΦ(Ω), TΦ[f ](x) = ∫ Ω Φ(x, y)f(y)dy ∀x ∈ Ω and provides a connection with the continuous basis that arises from an eigen-decomposition of TΦ. Finally, using the eigenvalues of this operator, we provide convergence estimates and stability bounds for interpolation and discrete least-squares approximation.