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

TL;DR

The paper introduces an adaptive Padé-Chebyshev approximation method for piecewise smooth functions that reduces Gibbs phenomenon and accurately captures singularities without prior knowledge, with proven error bounds and demonstrated efficiency.

Contribution

It presents a novel PiPCT approximation technique and an adaptive version (APiPCT) that improve accuracy and computational efficiency in approximating piecewise smooth functions.

Findings

PiPCT effectively minimizes Gibbs phenomenon.

APiPCT achieves desired accuracy with less computation.

Numerical results outperform existing methods.

Abstract

A piecewise Pad\'e-Chebyshev type (PiPCT) approximation method is proposed to minimize the Gibbs phenomenon in approximating piecewise smooth functions. A theorem on $L^1$-error estimate is proved for sufficiently smooth functions using a decay property of the Chebyshev coefficients. Numerical experiments are performed to show that the PiPCT method accurately captures isolated singularities of a function without using the positions and the types of singularities. Further, an adaptive partition approach to the PiPCT method is developed (referred to as the APiPCT method) to achieve the required accuracy with a lesser computational cost. Numerical experiments are performed to show some advantages of using the PiPCT and APiPCT methods compared to some well-known methods in the literature.