Kernel-based adaptive approximation of functions with discontinuities

One of the basic principles of Approximation Theory is that the quality of approximations increase with the smoothness of the function to be approximated. Functions that are smooth in certain subdomains will have good approximations in those subdomains, and these sub-approximations can possibly be calculated efficiently in parallel, as long as the subdomains do not overlap. This paper proposes an algorithm that first calculates sub-approximations on non-overlapping subdomains, then extends the subdomains as much as possible and finally produces a global solution on the given domain by letting the subdomains fill the whole domain. Consequently, there will be no Gibbs phenomenon along the boundaries of the subdomains. The method detects faults and gradient faults with good accuracy. Throughout, the algorithm works for fixed scattered input data of the function itself, not on spectral data, and it does not resample.