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Chebyshev interpolant at Gauss gridpoints

We consider the Chebyshev-Fourier expansion, (app_cheb.18)

We want to collocate the Chebyshev-Fourier coefficients at the Gauss quadrature points. Here we invoke the

. 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.

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