If $f$ is a $C^1$ function of period $2\pi$ and $f(0)=0$, and $g(x) = f(x)/(e^{ix} -1)$,
then let $C_n$ be the complex fourier coefficients of $f(x)$ and $D_n$ the coefficients of $g(x)$. How can we show that $D_n \to 0$?
ii) And how can we show that $C_n = D_{n-1} - D_n$, so that the series $\sum C_n$ is telescoping.
So, for the first one, I'm thinking of uniform continuity and how to show that $g(x)$ is continuous in it.