The starting point is the Gauss-Lobatto quadrature rule.
We make a short intermezzo on this issue.
If is an -orthogonal family of k-degree
polynomials, then by utilizing
Jacobi equation (2.4.9),
one finds that is k-degree family which is orthogonal
with respect to the weight . Applying Gauss rule
to the latter we find that there exist discrete gauss weights
such that
This is in fact a special case of the Gauss-Lobatto-Jacobi quadrature rule
which is exact for all . Indeed, all such p's
can be expressed as
with r(x) in ,
and a linear .
The last equality tells us that
Thus, we have
and the two expressions, II + III, amount to a linear combination of
p(-1) and p(1),
We conclude with
.
Let be an orthogonal family of k-degree
polynomials
in ,
where with
Let be the N+2
extrema of
. Then, there exist positive weights
such
that
. The Gauss-Lobatto-Chebyshev quadrature rule (corresponding
to and )
is nothing but
the familiar trapezoidal rule -- indeed
starting with (app_cheb.18), we have
and we end up with the discrete Chebyshev coefficients
This corresponds to the Fourier interpolant
with an even number of equidistant gridpoints
(consult (Fourier_even.2)), for
Then one may construct the Chebyshev interpolant at these N+1 gridpoints
We have an identical aliasing relation (compare (Fourier_even.5)),
(Verification: insert the Chebyshev expansion evaluated at into
(app_cheb.31),
to calculate the summation on the right we employ the
identity which yields
and (app_cheb.33) follows.)
The spectral Chebyshev estimate (app_cheb.28) together with
the aliasing relation (app_cheb.33)
yield the dospectral convergence estimate,
(compare (app_ps.17))
where .
: We have the Sobolev embedding of
with ,
Consequently,
In particular, with s=N+1 we obtain an improved estimate
for the near
min-max approximation collocated at ,