Theorem 22. Let be a weight function on the interval , as introduced in Eq. (9.17), and denote by the discrete truncated expansion of corresponding to Gauss, Gauss–Lobatto or Gauss–Radau quadratures. Then,That is, the discrete truncated expansion in orthogonal polynomials of is exactly equivalent to the interpolation of at the Gauss, Gauss–Lobatto or Gauss–Radau points.
The above simple proof did not assume any special properties of the polynomial basis, but does not hold for the Gauss–Lobatto case (for which the associated quadrature is exact for polynomials of degree ). However, the result still holds (at least for Jacobi polynomials); see, for example, .
Examples of Gauss-type nodal points are those given in Eq. (9.68) or Eq. (9.70). As we will see below, the identity (9.80) is very useful for spectral differentiation and collocation methods, among other things, since one can equivalently operate with the interpolant, which only requires knowledge of the function at the nodes.
Living Rev. Relativity 15, (2012), 9
This work is licensed under a Creative Commons License.