Eliminating oscillation in partial sum approximation of periodic function

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.