There is a paper of R. Bojanic and S.M. Mazhar, An estimate of the rate of
convergence of the Norlund.-Voronoi means of the Fourier series of functions
of bounded variation, Approx. Theory III, Academic Press (1980), 243-248
It says that if
$f:[-2\pi,2\pi]\rightarrow\mathbb{R}$ is $2\pi$-periodic and of bounded
variation and $S_{n}(f,x)$ is the partial sum of its Fourier series, then
$$
\left\vert \frac{1}{n}\sum_{k=1}^{n}S_{k}(f,x)-\frac{1}{2}(f_{+}%
(x)+f_{-}(x))\right\vert \leq\frac{c}{n}\sum_{k=1}^{n}\operatorname*{Var}%
\nolimits_{[0,\frac{\pi}{k}]}g_{x},
$$
where for every fixed $x\in\lbrack-2\pi,2\pi]$, $g_{x}(t):=f(x+t)+f(x-t)-f_{+}%
(x)-f_{-}(x)$ for $t\neq0$ and $g_{x}(0):=0$. Here $f_{+}(x)$ and $f_{-}(x)$
are the left and right limits.
In particular, if $f$ is piecewise $C^{1}$,
then
\begin{align*}
\operatorname*{Var}\nolimits_{\lbrack0,\frac{\pi}{k}]}g_{x} & =\int%
_{0}^{\frac{\pi}{k}}|g_{x}^{\prime}(t)|\,dt=\int_{0}^{\frac{\pi}{k}}%
|f^{\prime}(x+t)-f^{\prime}(x-t)|\,dt\\
& \simeq|f_{+}^{\prime}(x)-f_{-}^{\prime}(x)|\frac{1}{k}%
\end{align*}
and so
$$
\frac{c}{n}\sum_{k=1}^{n}\operatorname*{Var}\nolimits_{[0,\frac{\pi}{k}]}%
g_{x}\simeq|f_{+}^{\prime}(x)-f_{-}^{\prime}(x)|\frac{c}{n}\sum_{k=1}^{n}%
\frac{1}{k}\simeq|f_{+}^{\prime}(x)-f_{-}^{\prime}(x)|\frac{\log n}{n}.
$$
If $f_{+}^{\prime}(x)=f_{-}^{\prime}(x)$ and $f$ is piecewice $C^{2}$, then
$$
\int_{0}^{\frac{\pi}{k}}|f^{\prime}(x+t)-f^{\prime}(x-t)|\,dt\simeq
|f_{+}^{\prime\prime}(x)-f_{-}^{\prime\prime}(x)|\frac{1}{k^{2}}%
$$
and so
$$
\frac{c}{n}\sum_{k=1}^{n}\operatorname*{Var}\nolimits_{[0,\frac{\pi}{k}]}%
g_{x}\simeq|f_{+}^{\prime\prime}(x)-f_{-}^{\prime\prime}(x)|\frac{c}{n}%
\sum_{k=1}^{n}\frac{1}{k^{2}}.
$$
