Adaptive Pad\'e-Chebyshev Type Approximation to Piecewise Smooth Functions

TL;DR

This paper introduces adaptive piecewise Padé-Chebyshev approximation methods that effectively reduce Gibbs phenomena in approximating piecewise smooth functions without prior knowledge of singularities, validated by numerical experiments.

Contribution

The paper proposes novel adaptive piecewise Padé-Chebyshev algorithms that improve approximation accuracy and efficiency for piecewise smooth functions without needing prior singularity information.

Findings

The PiPCT algorithm minimizes Gibbs phenomena effectively.

The APiPCT algorithm achieves desired accuracy with lower computational cost.

Numerical results align well with theoretical error estimates.

Abstract

The aim of this article is to study the role of piecewise implementation of Pad\'e-Chebyshev type approximation in minimising Gibbs phenomena in approximating piecewise smooth functions. A piecewise Pad\'e-Chebyshev type (PiPCT) algorithm is proposed and an $L^1$-error estimate for at most continuous functions is obtained using a decay property of the Chebyshev coefficients. An advantage of the PiPCT approximation is that we do not need to have an {\it a prior} knowledge of the positions and the types of singularities present in the function. Further, an adaptive piecewise Pad\'e-Chebyshev type (APiPCT) algorithm is proposed in order to get the essential accuracy with a relatively lower computational cost. Numerical experiments are performed to validate the algorithms. The numerical results are also found to be well in agreement with the theoretical results. Comparison results of the PiPCT approximation with the singular Pad\'e-Chebyshev and the robust Pad\'e-Chebyshev methods are also presented.