6.4 Explicit verification for small N
To build intuition, we verify the theorem explicitly for small values of \(N\).
The rotated roots at \(\theta = 0\) are \(r_k = \cos (2\pi k/3)\) for \(k = 0, 1, 2\):
\begin{equation} r_0 = 1, \quad r_1 = \cos (2\pi /3) = -\frac{1}{2}, \quad r_2 = \cos (4\pi /3) = -\frac{1}{2}. \end{equation}
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The unscaled polynomial is
\begin{equation} P_3(x; 0) = (x - 1)\left(x + \frac{1}{2}\right)^2 = x^3 - \frac{3x}{4} - \frac{1}{4}. \end{equation}
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Scaling by \(2^{N-1} = 4\) gives
\begin{equation} S_3(x; 0) = 4x^3 - 3x - 1. \end{equation}
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This differs from \(T_3(x) = 4x^3 - 3x\) by the constant \(c(0) = -1\), as predicted.
The rotated roots at \(\theta = 0\) are \(r_k = \cos (\pi k/2)\) for \(k = 0, 1, 2, 3\):
\begin{equation} r_0 = 1, \quad r_1 = 0, \quad r_2 = -1, \quad r_3 = 0. \end{equation}
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The unscaled polynomial is
\begin{equation} P_4(x; 0) = (x - 1)(x + 1) x^2 = x^4 - x^2. \end{equation}
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Scaling by \(2^3 = 8\) gives
\begin{equation} S_4(x; 0) = 8x^4 - 8x^2. \end{equation}
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This differs from \(T_4(x) = 8x^4 - 8x^2 + 1\) by the constant \(c(0) = -1\).