Consider the following Fourier Series: $$ x^3- \pi^2 x = 12 \sum _{ n=1}^ { \infty} \frac { (-1)^n \sin(nx)} {n^3}, x \in (- \pi,\pi) $$
Can this Fourier Series be differentiated termwise?
The only result I know regarding termwise differentiation states that
If $f$ is periodic with period $2l$, continuous on $\mathbb R$ and $f’$ is piecewise continuous on $[-l,l]$ then the Fourier Series can be differentiated termwise.
So I thought of choosing $f(x)= x^3- \pi^2 x$ on $(-\pi,\pi)$ and extend it periodically on $\mathbb R$ but then $f$ won’t be continuous on real line and we cannot apply the result here. Am I missing something?
You can apply the well known theorem about differentiating sequences of functions, at least on each compact subinterval, since the sequence itself as well as the sequence of derivatives converges quite obviously uniformly on each compact subinterval of $(-\pi,\pi)$.
Edit: acutally $f$ and $f^\prime$, when extended periodically, are both continuous, so the theorem you cited does apply, as well.