1.2 Computational illustration: N = 5
To make the construction concrete, consider \(N = 5\) with \(\theta = 0\). The five roots of unity are
\begin{equation} \omega _k = \mathrm{e}^{2\pi i k/5}, \quad k = 0, 1, 2, 3, 4, \end{equation}
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positioned at angles \(0°, 72°, 144°, 216°, 288°\) on the unit circle. Their real projections are
\begin{equation} r_k(0) = \cos (2\pi k/5) = \{ 1, 0.309, -0.809, -0.809, 0.309\} . \end{equation}
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The monic polynomial with these roots is
\begin{equation} P_5(x; 0) = (x - 1)(x - 0.309)(x + 0.809)^2(x - 0.309), \end{equation}
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which simplifies to a polynomial of degree 5. Scaling by \(2^4 = 16\) gives
\begin{equation} S_5(x; 0) = 16x^5 - 20x^3 + 5x + c, \end{equation}
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where the coefficients of \(x^5, x^3, x\) match those of \(T_5(x) = 16x^5 - 20x^3 + 5x\), differing only in the constant term.
Rotating by \(\theta = \pi /10\) shifts the roots but preserves the polynomial coefficients (except the constant), demonstrating the invariance principle.