Approximation with rational and rescaled kernels: new steps forward

The aim of this paper is to present new insights into Rational Radial Basis Function (RRBF) approximations and their localized rescaled counterparts. After some necessary notation, we briefly recall the essence of stable computational techniques for overcoming the ill-conditioning of the kernel matrices, as well as the ones that explore stable bases. RRBFs are then introduced, and their benefits are described. We discuss Variably Scaled Kernels, which provide a more flexible tool for approximating with RBFs, substituting the shape parameter with a scaling function. This scaling function is then applied to the RRBFs to obtain a new family of RBFs that are more appropriate for discontinuous functions. Finally, the localized version of the RRBFs is recalled, for which we provide new insights into an open conjecture regarding the sum of the cardinal function associated with the approximant in rescaled form.