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### Chebyshev interpolant at Gauss-Lobatto gridpoints

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 ,    Next: Exponential convergence of Chebyshev Up: The Non-Periodic Problem -- Previous: Chebyshev interpolant at Gauss