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),