(Bartle's book; The elements of real analysis, 2nd edition, Exercise 38.u)
Suppose that $f$ and $f'$ are continuous with period $2\pi$ and that $f''$ is piecewise continuous with period $2\pi$. Show that the Fourier coefficients $a_n$, $b_n$ of $f$ are such that the series $\sum_{n=1}^{\infty} n^2 (|a_n| + |b_n|)$ is convergent.