. Let be an orthogonal
family of k-degree
polynomials
in ,
where with
.
Let be the N zeros of
. Then, there exist positive weights,
such that for all polynomials p(x) of degree we have
. To compute the Gauss weights we set
in (Gauss.rule).
Since ,
(Gauss.rule) yields
Equivalently, the corresponding weights are given by
To verify (Gauss.rule) we express p(x) as
for some (N-1)-degree
polynomials, t(x) and r(x). The choice of weights in (2.5.5)
guarantees that (Gauss.rule)
is valid for all polynomials of degree,
since the latter are spanned by .
This, together with the fact that is -orthogonal
to all polynomials of degree, implies
. The N-degree Gauss-Chebyshev
quadrature rule (based on the N+1 collocation points,
)
reads
with an error term, , which
vanishes for all polynomials of degree.
Applying the latter to the Fourier-Chebyshev coefficients in (cheb_gauss.1)
we arrive at discrete Chebyshev coefficients, which yield
We claim that is the N-degree algebraic interpolant of
w(x) at Chebyshev points . To see this we employ the
. There holds
We omit the straightforward proof of the general case
(-- which is based on the three step
recurrence relations for orthogonal polynomials), and concentrate
on the Chebyshev expansion in which case Christoffel-Darboux formula reads
Using this we find that interpolates w(x) at Chebyshev
points as asserted. Indeed we have
We want to estimate the error between w(x) and its Chebyshev interpolant
. As in the periodic Fourier case, we use here the
aliasing relation
which follows from the straightforward computation.
One concludes that the aliasing errors are dominated by
the spectrally small truncation error (app_cheb.28), and spectral
convergence follows.