Remarks on Fourier series

We prove the following propositions. An even integrable function whose Fourier coefficients form a convex sequence is absolutely continuous if and only if its Fourier series converges absolutely. If the function $f(t)$ is convex on $[0,\,\pi]$, $f(t)=f(\pi-t)$, then for odd $n$ $b_n=\frac2\pi\int_0^\pi f(t)\sin nt dt=\frac4\pi\frac{f(\pi/n)}n+\gamma_n$, $\sum_{n>1}|\gamma_n|<10\lceil f(\pi/2)\rceil$ while for even $n$, $b_n=0$.