Let \((V,\langle\cdot, \cdotangle)\) be a \(\mathbb{C}\)-inner product space, \(n \in \mathbb{N}\) and set \(\theta:=e^{2 \pi i / n}\).

(i) Show that

\[\frac{1}{n} \sum_{j=1}^{n} \theta^{j k}= \begin{cases}1 & \text { if } k=0 \\ 0 & \text { if } 1 \leqslant k \leqslant n-1\end{cases}\]

(ii) Use (i) to prove for all \(n \geqslant 3\) the following generalization of (26.5) and (26.6):

\[\langle v, wangle=\frac{1}{n} \sum_{j=1}^{n} \theta^{j}\left\|v+\theta^{j} wight\|^{2}\]

(iii) Prove the following continuous version of (ii):
\[
\langle v, wangle=\frac{1}{2 \pi} \int_{(-\pi, \pi]} e^{i \phi}\left\|v+e^{i \phi} wight\|^{2} d \phi
\]