Differentiation of a Fourier Series

Question

Fourier series for $f(x)=x$ where ($ -\pi< x< \pi $). Is it differentiable everywhere? Will the Fourier series be differentiable at all points ?

Answer 1

The Fourier series for $f(x)=x$ on $[-\pi,\pi]$ is \begin{align} S_f(x) & = \sum_{n=-\infty}^{\infty}\frac{1}{2\pi}\int_{-\pi}^{\pi}ye^{-iny}dy\,e^{inx} \\ & = \sum_{n=-\infty,\; n\mbox{ odd }}^{\infty}\left.\frac{1}{2\pi}y\frac{e^{-iny}}{-in}\right|_{y=-\pi}^{\pi} e^{inx} \\ & = \sum_{n=-\infty,\; n\mbox{ odd }}^{\infty}\frac{1}{in}e^{inx} \end{align} The series converges pointwise to $x$ on $(-\pi,\pi)$ and converges to $0$ at $\pm\pi$. So the Fourier series converges to a function that is differentiable everywhere except at $\pm\pi,\pm 3\pi,\pm 5\pi,\cdots$. The series even converges uniformly on every closed subinterval of $(-\pi,\pi)$. However, you cannot form the derivative of $S_f(x)$ by differentiating the Fourier series term by term, as the derived series does not converge because its general term does not tend to $0$.