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,
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,
[237
].
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.
http://www.livingreviews.org/lrr-2012-9 |
Living Rev. Relativity 15, (2012), 9
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