For some continuous function $f$ with period $2\pi$ on $\mathbb{R}$ with fourier series $\sum_{n\in\mathbb{Z}}c_ne^{inx}$, I had to show the following if and only if statement:
The function $f$ is infinitely differentiable if and only if there exists an $M>0$ such that $|c_n||n|^k\leq M$ for all $n$ and $k$, i.e. the corresponding fourier series converges uniformly (I think.)
I was able to show the forwards argument by inductively finding the fourier coefficients of the $k$th derivative of $f$ as $(-in)^kc_n$ by using integration by parts on the fourier series, which from here was not difficult to show that this is bounded. However I am having trouble showing that the second condition implies that $f$ is infinitely differentiable.
This part of the problem was provided with the hint that the following theorem should be used: Suppose $\{f_n\}$ is a sequence of functions, differentiable on $[a,b]$ and such that $\{f_n(x_0)\}$ converges for some point $x_0$ on $[a,b]$. If $\{f′n\}$ converges uniformly on $[a,b]$, then $\{f_n\}$ converges uniformly on $[a,b]$, to a function f, and $f′(x)=lim_{n→∞}f′_n(x)$.
It appears to me that this theorem is only applicable when we have information about $f'$ or subsequent derivatives of $f$, however we are trying to show that these exist, so I am uncertain how to properly apply this. I would appreciate it if somebody could outline a functioning proof, as well as explain where to apply the above theorem.