The rate of convergence of Cesaro average of Fourier series Do you know any estimates of rate of convergence of Cesaro average of Fourier series? It does not matter for which classes of functions. It would be great if you can give some estimates depending on the smoothness of the function.
It is well known that Cesaro average convergence uniformly for all continuous functions. Also for example there well known estimate for Fourier series (not Cesaro average) that looks like $O(\frac{\log n}{n^p})$ where $p$ is smoothness of the functions. I would like to know some analogical results for Cesaro sums.
Great thanks for any links, papers, books and so on!
 A: 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
$$
\frac{c}{n}\sum_{k=1}^{n}\operatorname*{Var}\nolimits_{[0,\frac{\pi}{k}]}%
g_{x}\simeq|f_{+}^{\prime\prime}(x)-f_{-}^{\prime\prime}(x)|\frac{c}{n}%
\sum_{k=1}^{n}\frac{1}{k^{2}}.
$$
