Showing sum of Fourier Coefficients converges for $f' \in L^2$.

Question

This is a practice problem for an applied analysis qualifying exam:

Let $f$ be a differentiable $2\pi$-periodic function on $\mathbb{R}$ with derivative $f' \in L^2([-\pi,\pi])$. Let $\{c_n\}_{n \in \mathbb{Z}}$ be the Fourier coefficient of $f$ with respect to the system $\{e^{inx}\}$.Prove that $$\sum_{n \in \mathbb{Z}} |c_n|<\infty.$$

I'm guessing that we need to justify term by term differentiation, then work with the Fourier Series for $f'$. I'm confused about when we may differentiate term by term in this case--all the references I've found seem to involve a lot of hand waving...

Answer 1

Just recall that the Fourier coefficients of $f$ are given by $$c_n=\int_0^{2\pi}\,f(t)e^{-int}\,dt$$ and those of $f'$ by $$d_n=\int_0^{2\pi}\,f'(t)e^{-int}\,dt\text{.}$$ Integrating by parts (with $u=f$ and $v=\frac{e^{-int}}{-in}$), for $n\neq 0$, $$c_n=\left[f(t)\frac{e^{-int}}{-in}\right]_0^{2\pi}+\frac{1}{in}\int_0^{2\pi}\,f'(t)e^{-int}\,dt=\frac{d_n}{in}$$ as $f(t)\frac{e^{-int}}{-in}$ is periodic of period $2\pi$. In this way, we have that $$\sum_{n\neq 0}\,|c_n|=\sum_{n\neq 0}\frac{|d_n|}{n}\leq \sqrt{\sum_{n\neq 0}\,|d_n|^2}\sqrt{\sum_{n\neq 0}\frac{1}{n^2}}<\infty\text{,}$$ by the Cauchy-Schwarz inequality, and so $\sum_{n=-\infty}^\infty\,|c_n|$ is finite as desired.

The only important part of this argument is noting that manipulating the integral formula of the Fourier coefficients rigorously is easier than manipulating them in the series directly as there are more subtleties in the convergence of the series, i.e., the meaning of $f(t)=\sum_n\,c_ne^{int}$.