I'd like to to prove the following statement:
For every $N \geq 1$, there exists $C>0$ such that $|D_N(t)| \leq C|t|^{-1}$, for $|t|<\frac{1}{2}$, where $D_N(t) = \sum_{k=-N}^N e^{2\pi ikt} = \frac{\sin{((2N+1) \pi x)}}{\sin(\pi x)}$ and $\{ D_N \}$ is the Dirichlet kernel.
Not sure how to approach this, but here is my work:
Since $\frac{\pi t}{\sin(\pi t)} \to 1$, there exists some $\delta > 0$ such that $\left| \frac{\pi t}{\sin (\pi t)} - 1 \right| < 1$ whenever $|t| < \delta$. From here we get $0 < \frac{\pi t}{\sin (\pi t)} < 2$, i.e., $\frac{1}{|\sin(\pi t)|} \leq \frac{2}{\pi} \frac{1}{|t|}$. It follows that $|D_N(t)| \leq \frac{2}{\pi}\frac{1}{|t|}$. This is almost what I want, except that if $\delta < \frac{1}{2}$, I don't know how to cover the $\delta < |t| < \frac{1}{2}$ case.
Thanks in advance for any help.
