I am given the following sequence of functions $(f_m)_{m \in \mathbb{N}}$. They are defined by $$ f_m(x):=\left( \frac{e^{-ix}-1}{-ix} \right)^m \left( \sum_{l \in \mathbb{Z}} \frac{\left|e^{-ix}-1 \right|^{2m}}{|x+ 2 \pi l |^{2m}} \right)^{-\frac{1}{2}}$$ for $x \neq 2 \pi l$ with $l \in \mathbb{Z}$ and $f_m(2 \pi l)=1$. Now, I want to show that this sequence converges to the characteristic function $\chi_{[-\pi,\pi]}$ in the $L^2$ sense?
So $\|f_m- \chi_{[-\pi,\pi]} \|_2 \rightarrow 0.$
The idea could be to use the dominated convergence theorem. For this we need to find a uniform upper bound and be sure that pointwise convergence is satisfied.
Regarding the upper bound it may be useful to see that
$$|f_m(x)|^2 = \left( \sum_{l \in \mathbb{Z}} \frac{\left|x \right|^{2m}}{|x+ 2 \pi l |^{2m}} \right)^{-1}.$$