Piecewise nonlinear approximation for non-smooth functions

Abstract

Piecewise affine or linear approximation has garnered significant attention as a technique for approximating piecewise-smooth functions. In this study, we propose a novel approach: piecewise non-linear approximation based on rational approximation, aimed at approximating non-smooth functions. We introduce a method termed piecewise Padé Chebyshev (PiPC) tailored for approximating univariate piecewise smooth functions. Our investigation focuses on assessing the effectiveness of PiPC in mitigating the Gibbs phenomenon during the approximation of piecewise smooth functions. Additionally, we provide error estimates and convergence results of PiPC for non-smooth functions. Notably, our technique excels in capturing singularities, if present, within the function with minimal Gibbs oscillations, without necessitating the explicit specification of singularity locations. To the best of our knowledge, prior research has not explored the use of piecewise non-linear approximation for approximating non-smooth functions. Finally, we validate the efficacy of our methods through numerical experiments, employing PiPC to reconstruct a non-trivial non-smooth function, thus demonstrating its capability to significantly alleviate the Gibbs phenomenon.