6.1 Chebyshev roots and their power sums
To prove that \(S_N(x; \theta ) = T_N(x) + c(\theta )\), we must show that the roots \(r_k(\theta )\) and the Chebyshev roots \(\xi _k\) have matching power sums. We already know the rotated roots have \(\theta \)-independent power sums. Now we compute the power sums of the Chebyshev roots explicitly.
The roots of the Chebyshev polynomial \(T_N(x)\) are
These are \(N\) distinct real numbers in the interval \((-1, 1)\).
By the defining relation \(T_N(\cos \theta ) = \cos (N\theta )\), we have \(T_N(\xi _k) = 0\) if and only if \(N\theta _k = (2k + 1)\pi /2\) for some integer \(k\), i.e., \(\theta _k = (2k + 1)\pi /(2N)\). Since \(\theta _k \in (0, \pi )\) for \(k \in \{ 0, \ldots , N-1\} \), the values \(\xi _k = \cos \theta _k\) are distinct and lie in \((-1, 1)\).