Chebyshev–Gauss–Lobatto collocation grid

Let $N\ge 1$ be an integer and put $x_j=\cos (j\pi \mathrm{/}N)$ for $j=0,\dots ,N$. The ordering is decreasing: $x_0=1$ and $x_N=-1$. These are the Chebyshev–Gauss–Lobatto nodes; $T_n(\cos \theta )=\cos (n\theta )$ and $U_n(\cos \theta )\sin \theta =\sin ((n+1)\theta )$ for every $n\in \mathbb{N}$ gives $T_N(x_j)=(-1{)}^j$. They are distinct because cosine is strictly decreasing on $[0,\pi ]$ by Signs, monotonicity intervals, and ranges of sine and cosine.

For interpolation using Cardinal-polynomial interpolation, use the increasing reindexing $z_k=x_{N-k}$, $k=0,\dots ,N$. For descending-grid real data $U_j$, the corresponding increasing data are $Y_k=U_{N-k}$.

For real nodal data $U_0,\dots ,U_N\in \mathbb{R}$, define $I_N^{L}U$ to be the cardinal interpolant of $(Y_k{)}_{k=0}^N$ at $(z_k{)}_{k=0}^N$. For a function $f:[-1,1]\to \mathbb{R}$, write $I_N^{L}f$ for the interpolant of its nodal values. The increasing-node interpolation theorem Existence and uniqueness of the cardinal-polynomial interpolant makes this polynomial unique.