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 ,

Thu Jan 22 19:07:34 PST 1998