4.2.2 Example: \(N = 4\), \(j = 3\)
For \(N = 4\) and \(j = 3\):
\begin{equation} S_3(\theta ) = \sum _{k=0}^3 \cos ^3\left(\theta + \frac{\pi k}{2}\right). \end{equation}
5
Using the identity \(\cos ^3\theta = \frac{3\cos \theta + \cos (3\theta )}{4}\):
\begin{equation} S_3(\theta ) = \frac{3}{4}\sum _{k=0}^3 \cos \left(\theta + \frac{\pi k}{2}\right) + \frac{1}{4}\sum _{k=0}^3 \cos \left(3\theta + \frac{3\pi k}{2}\right). \end{equation}
6
Both sums vanish (first sum by Lemma 3.1.1, second sum by Lemma 3.1.2 with \(m = 3 {\lt} N = 4\)), giving \(S_3(\theta ) = 0\) for all \(\theta \).