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Spectral accuracy

In order to measure the spectral convergence of Chebyshev expansion, we have to estimate the decay rate of Chebyshev coefficients in terms of the smoothness of w(x) and its derivatives; to this end we need Sobolev like norms. Unlike the Fourier case, is not complete with respect to - orthogonality is lost because of the Chebyshev weight. So we can proceed formally as before, see (app_fourier.24),

i.e., if we define the Chebyshev-Sobolev norm

then we have spectral accuracy

In fact the space can be derived from an appropriate inner product in the real space as done in Fourier expansion. The correct inner product -- expressed in terms of , is given by (in analogous manner to (app_fourier.19))

so that

Hence the Fourier coefficients in this Hilbert space behave like

and the corresponding norm is equivalent to

The reason for the squared factors here is due to the fact that L is a second order differential operator, unlike the first-order in the Fourier case, i.e.,

involves the first 2s-derivatives of w(x) - appropriately weighted by Chebyshev weight. This completes the analogy with the Fourier case, and enables us to estimate derivative as well-compare (app_fourier.28),