Mapped polynomials and discontinuous kernels for Runge and Gibbs phenomena

In this paper, we present recent solutions to the problem of approximating functions by polynomials for reducing in a substantial manner two well-known phenomena: Runge and Gibbs. The main idea is to avoid resampling the function or data and relies on the mapped polynomials or “fake” nodes approach. This technique turns out to be effective for stability by reducing the Runge effect and also in the presence of discontinuities by almost cancelling the Gibbs phenomenon. The technique is very general and can be applied to any approximant of the underlying function to be reconstructed: polynomials, rational functions or any other basis. A “natural” application is then quadrature, that we started recently to investigate and we propose here some results. In the case of jumps or discontinuities, where the Gibbs phenomenon appears, we propose a different approach inspired by approximating functions by kernels, in particular Radial Basis Functions (RBF). We use the so called Variably Scaled Discontinuous Kernels (VSDK) as an extension of the Variably Scaled Kernels (VSK) firstly introduced in Bozzini et al. (IMA J Numer Anal 35:199–219, 2015). VSDK show to be a very flexible tool suitable to substantially reducing the Gibbs phenomenon in reconstructing functions with jumps. As an interesting application we apply VSDK in Magnetic Particle Imaging which is a recent non-invasive tomographic technique that detects super-paramagnetic nanoparticle tracers and finds applications in diagnostic imaging and material science. In fact, the image generated by the MPI scanners are usually discontinuous and sampled at scattered data points, making the reconstruction problem affected by the Gibbs phenomenon. We show that VSDK are well suited in MPI image reconstruction also for identifying image discontinuities.