Adaptive Approximation of Functions with Discontinuities
This addresses the problem of accurate approximation of discontinuous functions for applications in scientific computing and signal processing, offering a novel approach to avoid Gibbs artifacts.
The paper proposes a class of algorithms for approximating functions with discontinuities by computing sub-approximations on non-overlapping subdomains and extending them to cover the whole domain, eliminating the Gibbs phenomenon. The method works with fixed scattered data and does not require resampling.
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 {\em sub-approximations} can possibly be calculated efficiently in parallel, as long as the subdomains do not overlap. This paper proposes a class of algorithms that first calculate sub-approximations on non-overlapping subdomains, then extend the subdomains as much as possible and finally produce 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. Throughout, the algorithm works for fixed scattered input data of the function itself, not on spectral data, and it does not resample.