4.1 Strategy via binomial expansion
The key technique is to expand \(\cos ^j(\theta + 2\pi k/N)\) using Euler’s formula:
\begin{equation} \cos \theta = \frac{\mathrm{e}^{i\theta } + \mathrm{e}^{-i\theta }}{2}. \end{equation}
1
Raising to the \(j\)-th power and expanding binomially yields
\begin{equation} \cos ^j\theta = \frac{1}{2^j} \sum _{\ell =0}^j \binom {j}{\ell } \mathrm{e}^{i(j - 2\ell )\theta }. \end{equation}
2
Summing over \(k = 0, \ldots , N-1\) with \(\theta = \theta + 2\pi k/N\), each term with \(j - 2\ell \not\equiv 0 \pmod{N}\) contributes zero by discrete orthogonality (Lemma 3.1.2). Only the term with \(j - 2\ell \equiv 0 \pmod{N}\) survives, and this term is independent of \(\theta \).