2.1 Chebyshev polynomials
The Chebyshev polynomials of the first kind \(T_N(x)\) are defined by the relation
\begin{equation} T_N(\cos \theta ) = \cos (N\theta ). \end{equation}
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This uniquely determines \(T_N\) as a polynomial of degree \(N\). The first few Chebyshev polynomials are:
\begin{align*} T_0(x) & = 1, \\ T_1(x) & = x, \\ T_2(x) & = 2x^2 - 1, \\ T_3(x) & = 4x^3 - 3x, \\ T_4(x) & = 8x^4 - 8x^2 + 1. \end{align*}
The Chebyshev polynomials satisfy the following properties:
Recurrence: \(T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)\) for \(n \geq 1\).
Leading coefficient: The leading coefficient of \(T_N(x)\) is \(2^{N-1}\) for \(N \geq 1\).
Roots: The roots of \(T_N(x)\) are
\begin{equation} \label{eq:chebyshev_roots} x_k = \cos \left(\frac{(2k + 1)\pi }{2N}\right), \quad k = 0, 1, \ldots , N-1. \end{equation}2Extrema: On \([-1, 1]\), \(T_N\) attains its extremal values \(\pm 1\) at the \(N + 1\) points \(\cos (j\pi /N)\) for \(j = 0, 1, \ldots , N\).