On the Gibbs Phenomenon of Fourier Series of a Classical Function

Abstract:

      In this paper, we point out that the Fourier series of a classical function $\sum_{k=1}^{\infty}{\frac{\sin kx}k}$ has the Gibbs phenomenon in the neighborhood of zero. Furthermore, we estimate the upper bound of its partial sum and get: $$\sup_{n\geq 1} {\big\|\sum_{k=1}^n{\frac{\sin kx}k}\big\|=\int_0^{\pi}{\frac {\sin x}x\d x}}\doteq 1.85194,$$ which is better than that in [1].