It appears there are somewhat conflicting definitions of the points used in Chebyshev interpolation. Wikipedia and Numerical Recipes define the $x_j^{(n)}$ sample points for $(n-1)^\text{th}$-order Chebyshev interpolation as the zeroes of $T_n(x)$ (the $n^\text{th}$ Chebyshev polynomial) such that

$$ x_j^{(n)} = \cos\left(\frac{j + \frac{1}{2}}{n}\pi\right); \quad j = 0, \ldots, n - 1. $$

In his works on interpolation, however, Nick Trefethen defines the Chebyshev points by the extrema of $T_n(x)$,

$$ \tilde{x}_j^{(n)} = - \cos\left(\frac{j}{n} \pi\right); \quad j = 0, \ldots, n, $$ instead of the zeroes. Notably,

  1. The definition of $x_j^{(n)}$ will never include the endpoints of the $x \in [-1, 1]$ interval.
  2. An $n^\text{th}$-order interpolation conventionally has $n + 1$ terms. Using the $x_j^\text{(n)}$ for interpolation will only use the first $n$ Chebyshev polynomials (as $j = 0, \ldots, n-1$ has $n$ terms).

It seems, then, that Trefethen's $\tilde{x}_j^{(n)}$ are the superior sample points as I imagine it's quite helpful to include the endpoints of the interval in any interpolation beyond $0^\text{th}$ order and the $x_j^{(n)}$ choice doesn't even allow for $0^\text{th}$ order interpolation ($T_0(x) = 1$). So my questions are as follows:

  1. Are the extrema of the $n^\text{th}$ Chebyshev polynomial (Trefethen's convention) actually the superior choice for sampling and interpolation applications?
  2. How did it arise that there are multiple definitions for "Chebyshev sample points"?

Update: This paper (The Chebyshev points of the first kind by Kuan Xu) shows an analogue between what the author calls the "extrema grid" and the "nodal" grid, demonstrating they have many similar properties. The author does not make claim as to which one is "better," however, only that the Chebyshev points of the second kind (the extrema grid) are "much more common."


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.