Chapter 2: Preliminaries
Chebyshev polynomials
The Chebyshev polynomials of the first kind \(T_N(x)\) are defined by the relation
\begin{equation}...\end{equation}This uniquely determines \(T_N\) as a polynomial of degree \(N\). The first few Chebyshev polynomials are:
\begin{align*}...\end{align*}The Chebyshev polynomials satisfy the following properties:
Recurrence: \(T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)\) for \(n \geq 1\).
Polynomial.Chebyshev.T_add_twoLeading coefficient: The leading coefficient of \(T_N(x)\) is \(2^{N-1}\) for \(N \geq 1\).
chebyshev_T_leadingCoeffRoots: The roots of \(T_N(x)\) are \begin
equation
x_k = \cos\left(\frac
(2k + 1)\pi
2N
\right), \quad k = 0, 1, \ldots, N-1.
chebyshevRoot, chebyshevRootsList, chebyshev_T_eval_chebyshevRoot, chebyshev_T_eval_eq_zero_iff \item Extrema: On \([-1, 1]\), \(T_N\) attains its extremal values \(\pm 1\) at the \(N + 1\) points \(\cos(j\pi/N)\) for \(j = 0, 1, \ldots, N\).
2.1 Chebyshev polynomials
The Chebyshev polynomials of the first kind \(T_N(x)\) are defined by the relation
\begin{equation}...\end{equation}This uniquely determines \(T_N\) as a polynomial of degree \(N\). The first few Chebyshev polynomials are:
\begin{align*}...\end{align*}The Chebyshev polynomials satisfy the following properties:
Recurrence: \(T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)\) for \(n \geq 1\).
Polynomial.Chebyshev.T_add_twoLeading coefficient: The leading coefficient of \(T_N(x)\) is \(2^{N-1}\) for \(N \geq 1\).
chebyshev_T_leadingCoeffRoots: The roots of \(T_N(x)\) are \begin
equation
x_k = \cos\left(\frac
(2k + 1)\pi
2N
\right), \quad k = 0, 1, \ldots, N-1.
chebyshevRoot, chebyshevRootsList, chebyshev_T_eval_chebyshevRoot, chebyshev_T_eval_eq_zero_iff \item Extrema: On \([-1, 1]\), \(T_N\) attains its extremal values \(\pm 1\) at the \(N + 1\) points \(\cos(j\pi/N)\) for \(j = 0, 1, \ldots, N\).