Eliminating oscillation in partial sum approximation of periodic function

TL;DR

This paper proposes a new method to eliminate oscillations, such as the Gibbs phenomenon, in partial sum approximations of periodic functions, improving the accuracy of series-based function representations.

Contribution

The paper introduces an approach specifically designed to remove oscillations in partial sums of periodic functions, addressing a longstanding challenge in Fourier series approximation.

Findings

Effective reduction of Gibbs phenomenon oscillations

Improved accuracy in partial sum approximations

Applicable to various periodic functions

Abstract

If we cannot obtain all terms of a series, or if we cannot sum up a series, we have to turn to the partial sum approximation which approximate a function by the first several terms of the series. However, the partial sum approximation often does not work well for periodic functions. In the partial sum approximation of a periodic function, there exists an incorrect oscillation which cannot be eliminated by keeping more terms, especially at the domain endpoints. A famous example is the Gibbs phenomenon in the Fourier expansion. In the paper, we suggest an approach for eliminating such oscillations in the partial sum approximation of periodic functions.