Spectral differentiation

From chapter 9, instead of using local node set, use all of the nodes for global interpolation. The convergence is spectral if \(f\) has infinitely many derivatives on the interval (carries through to the derivatives).

Chebyshev differentiation matrix is a dense matrix with entries:

\[ D_{00} = \frac{2n^2 +1}6, \quad D_{nn} = -\frac{2n^2 +1}6\\ D_{ij} = \begin{cases} -\frac{x_i}{2(1-x^2_i)}, & i - j,\\ \frac{c_i}{c_j}\frac{(-1)^{i+j}}{x_i-x_j}, & i \neq j, \end{cases} \]