I'm working on the following problem:
Let $f$ be a continuously differentiable, $2\pi$-periodic function. Prove that
The Fourier series coefficients given by $$ c_n=\frac1{\sqrt{2\pi}}\int_0^{2\pi}f(x)e^{-inx}dx $$ satisfies $$ \sum_{n\in\mathbf Z}|c_n|^2(1+n^2)<\infty. $$
The Fourier series $\sum_{n\in\mathbf Z}c_ne^{-inx}$ converges uniformly to $f$.
If $f$ is in fact of class $C^k$ then the $m$-times differentiated Fourier series of $f$ converges uniformly to $f^{(m)}$.
For the first subproblem, I am asked to compute the Fourier series coefficients of $f'$: $$ d_n=\frac1{\sqrt{2\pi}}\int_0^{2\pi}f'(x)e^{-inx}dx. $$ Integrating by parts with $u=e^{-inx}dx$ and $dv=f'(x)dx$ gives $du=-ine^{inx}dx$ and $v=f(x)$, so that $$ d_n=\frac1{\sqrt{2\pi}}\left(f(x)e^{-inx}\Big|_0^{2\pi}+in\int_0^{2\pi}f(x)e^{inx}dx\right). $$ Because $f$ is $2\pi$-periodic, the first term vanishes, so that $$ d_n=inc_n. $$ However, I'm not quite sure how to proceed with showing that $\sum|c_n|^2(1+n^2)$ is finite. Any help or intuition would be appreciated!
Answer for (1): $\sum |d_n|^{2} <\infty$ so $\sum n^{2}|c_n|^{2} <\infty$. Note that $(1+n^{2})|c_n|^{2} \leq 2 n^{2}|c_n|^{2}$.
Answer for (2): $(\sum |c_n|)^{2} =(\sum |\frac 1 n (nc_n)|)^{2} \leq \sum n^{2}|c_n|^{2} \sum \frac 1 {n^{2}}$. Hence $\sum |c_n| <\infty$ and the series $\sum c_n e^{-incx}$ converges uniformly by M-test. (The sum of the Fourier series has to be $f$ since the partial sums converge in $L^{2}$ to $f$).