Chebyshev Nodes with Interval Endpoints

Question

The "classic" Chebyshev nodes on interval $[-1, 1]$ are given by

$$ x_k = \cos \left( \frac{2k - 1}{2n} \pi \right), k = 1, \dots, n. $$ These are the roots of the Chebyshev polynomials of first kind, and also partition a semicircle into equally spaced intervals.

Now my question: Is there literature out there considering Chebyshev nodes "with endpoints", i.e.,

$$ \widetilde{x}_k = \cos \left( \frac{k }{n} \pi \right), k = 0, \dots, n? $$

One can observe that these nodes also lead to points which seperate the circle into intervals of equal arc length:

For $n=6$:

For $n=7$:

Are there any special properties of the $\widetilde{x}_k$ (besides seperating the circle equally) known?

Answer 1

The points $\tilde{x}_k$'s are called Chebyshev–Lobatto points, Chebyshev extreme points, or Chebyshev points of the second kind. See e.g. Lloyd N. Trefethen, Approximation Theory and Approximation Practice (SIAM, 2012). Note that in this book the author calls $\tilde{x}_k$'s Chebyshev nodes, which I find non-standard.

Answer 2

They are called "Chebyshev Extreme Points" since

$$ \widetilde{x}_k = \cos \left( \frac{k }{n} \pi \right), k = 0, \dots, n $$ are the extrema of the $n$'th Chebyshev polynomial of first kind $T_n$ on the interval $x \in [-1, 1]$.