Editing Chebyshev nodes (section)

==Definition==
[[File:ChebyshevNodes2.jpg|thumb|Chebyshev nodes of both kinds from <math>n=2</math> to <math>n=50</math>.]]

For a given positive integer <math>n</math>, the {{tmath|n}} Chebyshev nodes of the first kind are given by

<math display="block">x_k = \cos\frac{\bigl(k+\tfrac12\bigr)\pi}{n}, \quad k = 0, \ldots, n-1.</math>

This is the projection of {{tmath|2n}} equispaced points on the unit circle onto the interval {{tmath|[-1, 1]}}, the circle's diameter. These points are also the roots of {{tmath|T_n}}, the Chebyshev polynomial of the first kind with degree {{tmath|n}}.

The {{tmath|n+1}} Chebyshev nodes of the second kind are given by

<math display="block">x_k = \cos\frac{k\pi}{n}, \quad k = 0, \ldots, n.</math>

This is also the projection of {{tmath|2n}} equispaced points on the unit circle onto {{tmath|[-1, 1]}}, this time including the endpoints of the interval, each of which is only the projection of one point on the circle rather than two. These points are also the extrema of {{tmath|T_n}} in {{tmath|[-1, 1] }}, the places where it takes the value {{tmath|\pm1}}.<ref>{{harvnb|Trefethen|2013|pp=14}}</ref> The interior points among the nodes, not including the endpoints, are also the zeros of {{tmath|U_{n-1} }}, a Chebyshev polynomial of the second kind, a rescaling of the derivative of {{tmath|T_n}}.

For nodes over an arbitrary interval <math>[a,b]</math> an [[affine transformation]] from <math>[-1,1]</math> can be used:
<math display="block">\tilde{x}_k = \tfrac12(a + b) + \tfrac12(b - a) x_k.</math>