Let a function $f: \mathbb{R} \rightarrow \mathbb{R}$ be integrable on $[-\pi,\pi]$ and $2\pi$-periodic. Let $ \frac{a_0}{2}+\sum\limits_{n=1}^\infty (a_n \cos nx+b_n \sin nx) $ be the Fourier series of $f$.
Assume that $f$ has continuous derivative at $a$. How to show that of partial sum $T_n$ of the series $ \sum\limits_{n=1}^\infty (-n a_n \sin nx+n b_n \cos nx) $ (it is the Fourier series of $f$ after termwise differentiation) is convergent in arithmetic mean in $a$ to $f'(a)$? That is $ \lim\limits_{n\to\infty}\sigma_n(x):=\lim\limits_{n\to\infty}\frac{T_0(a)+T_1(a)+\ldots +T_{n-1}(a)}{n-1} = f'(a) $ Thanks.