A new stable basis for radial basis function interpolation

• Authors: Stefano De Marchi;Gabriele Santin
• Venue: Journal of Computational and Applied Mathematics
• Year: 2013

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), [5], the authors gave a quite general way to build stable and orthonormal bases for the native space N"@F(@W) associated to a kernel @F on a domain @W@?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 N"@F(@W) and in @?"2","w(X), where X is a set of data sites of the domain @W. 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"@F:N"@F(@W)-N"@F(@W), T"@F[f](x)=@!"@W@F(x,y)f(y)dy@?x@?@W, and provides a connection with the continuous basis that arises from an eigendecomposition of T"@F. Finally, using the eigenvalues of this operator, we provide convergence estimates and stability bounds for interpolation and discrete least-squares approximation.