Fourier series of a even function

Question

We define $f(x)=\textbf{1}_{[-\pi/2,\pi/2]}(x)$, $x \in [-\pi,\pi]$. We consider a Fourier series of $f$. For $n \in \mathbb{Z}$, we have \begin{align*} \widehat{f}(n)&:=\frac{1}{2\pi}\int_{-\pi}^\pi f(x)e^{-inx}\,dx=\frac{1}{2\pi}\frac{\widehat{f}(n)+\widehat{f}(-n)}{2}\\ &=\frac{1}{2\pi}\int_{-\pi/2}^{\pi/2}\cos(nx)\,dx=\frac{1}{\pi n} \sin(n\pi/2). \end{align*} Therefore, the Fourier series $\sum_{n \in \mathbb{Z}}\widehat{f}(n)e^{inx}$ does not converge uniformly on $[-\pi,\pi]$. However, can we prove that this series converges for each $x \in [-\pi,\pi]$?

Answer 1

The short answer is no, Because you do not have pointwise convergence of the fourier series at points of discontinuity. This is called the Gibbs phenomenon.

https://en.wikipedia.org/wiki/Gibbs_phenomenon

You can find the formal description in the article, which can be directly applied to your function.