Let $f$ be an integrable function defined on $[-\pi,\pi]$. Our goal is to prove the following theorem(due to Dirichlet):
If $f$ is continuous in a neighborhood of $t_{0} \in [-\pi,\pi]$ and has right and left-sided limits at $t_{0}$ and if the left and right side generalized derivatives $$f_{R}^{'}(t_{0}) = \lim_{t\rightarrow t_{0}^{+}}\frac{f(t) - f(t_{0}^{+})}{t - t_{0}}, f_{L}^{'}(t_{0}) = \lim_{t\rightarrow t_{0}^{-}}\frac{f(t) - f(t_{0}^{-})}{t - t_{0}}$$
both exist, then $$\sum_{n \in \mathbb{Z}}c_{n}(f)e^{-nt_{0}} = \frac {f(t_{0}^{+}) + f(t_{0}^{-})}{2}. $$
(a)Write $S_{n}(t_{0}) = \sum_{k=-n}^{n} c_{n}(f)e^{-nt_{0}}$ as a convolution $S_{n}(t_{0}) = f*D_{n}(t_{0}).$ What is $D_{n}$?
I think the index $k$ in the summation is wrong as the summed value does not contain $k$ at all. am I right? why here in the partial sum definition the exponential does not contain the imaginary i, is it a mistake or no?