---- > [!definition] Definition. ([[Dirichlet Kernel]]) > The $N ^{th}$ **Dirichlet Kernel** is the function $D_{N}(\theta) := \sum_{n=-N}^{N} e ^{i n \theta } = \frac{\sin\left( ( N+\frac{1}{2} )\theta \right) }{\sin \left( \frac{\theta}{2} \right)}, \ \ \theta \in \rr.$ > [!justification] > We prove here the second equality above. Using the second [[finite geometric series]] expression with $w= e ^{in \theta}$, write $\begin{align} > D_{N}(\theta) = & \frac{e ^{-i N \theta} - e ^{-i(N+1)\theta}}{1 - e ^{i \theta}} \\ > = & \frac{e ^{-i N \theta} - e ^{-i(N+1)\theta}}{1 - e ^{i \theta}} \cdot \frac{e ^{-\frac{i \theta}{2}}}{e ^{-\frac{i \theta}{2}}} \\ > = & \frac{e ^{-i(N+\frac{1}{2})\theta} - e ^{i(N+ \frac{1}{2})\theta}}{e ^{-i\theta /2} + e ^{i \theta /2}} \\ > \end{align}$ > Now the result follows by applying [[Euler's formula]] to the top and bottom. > ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```