Fejer summation of the square wave

Direct integration gives $\widehat{q}(2m)=0$ for every $m$ and $\widehat{q}(2m+1)=\frac{2}{\pi i(2m+1)},\widehat{q}(-(2m+1))=-\frac{2}{\pi i(2m+1)}\mathrm{.}$ Hence $S_{j}q(x)=\frac{4}{\pi }{\sum }_{0\le m,2m+1\le j}\frac{\sin (2\pi (2m+1)x)}{2m+1}\mathrm{.}$

Averaging the partial sums from step 1.1 as in [L1] shows that the odd mode $2m+1$ appears with weight $1-(2m+1)\mathrm{/}(N+1)$ when $2m+1\le N$ and with weight $0$ otherwise. This is exactly the displayed formula for ${\sigma }_{N}q$.

The one-sided limits at $0$ are $-1$ and $1$, so [L2] gives ${\sigma }_{N}q(0)\to \frac{-1+1}{2}=0.$ Also [L1] and [L3] give ${\sigma }_{N}q(x)={\int }_0^{1}q(x-t)F_N(t)dt,$ and because $\mathrm{\mid }q\mathrm{\mid }\le 1$, positivity and total mass one imply $\mathrm{\mid }{\sigma }_{N}q(x)\mathrm{\mid }\le 1$ for every $x$. Thus the Fejer means average across the jump and do not exhibit a fixed overshoot.