I am studying the Fourier series in my real analysis course and I am stuck on some questions...
I have a function $f(x)$ in the trigonometric system given as:
$f(x)\sim \frac{a_0}2+\sum_{n=1}^\infty\bigl(a_n\cos nx + b_n\sin nx\bigr)$
and I want to show that the derivative of this function is:
$f'(x)\sim \sum_{n=1}^\infty\bigl(nb_n\cos nx -n a_n\sin nx\bigr)$
It can be proved easily using the definitions but my question is how do we know if we can expand f' in the trig system or not? Based on my current knowledge I think we need f' to be in $L^2$ so that the expansion can be considered.
There are some conditions given on $f$ but I can't find a way to prove that $f'$ is in $L^2$. The conditions given are that $f$ is $L^2[-\pi, \pi]$ and is $2\pi$ periodic and has a continuous derivative. Can you help me prove that $f'$ is also in $L^2$? I tried to use Jordan's test as well but can't see a way to prove that f' is of bounded variation either.