Cesàro means of Fourier series

Remarks

Counting how often each frequency occurs in $S_0 f, \ldots, S_{N-1} f$ gives $$\sigma_N f(x) = \sum_{\abs{k} < N} \left( 1 - \frac{\abs{k}}{N} \right) \hat f(k) e^{ikx} = \frac{1}{2\pi} \int_{-\pi}^{\pi} F_N(t) f(x - t) \, dt,$$ where $F_N$ is the Fejér kernel. So $\sigma_N f$ is a trigonometric polynomial of degree at most $N - 1$. It is the Cesàro mean of order $1$ of the sequence $(S_n f(x))_{n \ge 0}$.

Example: for $f(x) = \cos x$, $S_0 f = 0$ and $S_n f = \cos x$ for $n \ge 1$. So $\sigma_N f = \frac{N-1}{N} \cos x$, which tends to $f$.

Non-example: the Abel–Poisson means $\sum_k r^{\abs{k}} \hat f(k) e^{ikx}$ also average the Fourier series, but with the weights $r^{\abs{k}}$ instead of $1 - \abs{k}/N$; they are a different method, the Abel–Poisson summation.