We conclude with a discussion on Chebyshev differencing. Starting with
grid values at Chebyshev points , one constructs the Chebyshev interpolant
One can compute , efficiently via the cos-FFT
with operations. Next, we differentiate in
Chebyshev space
In this case, however, is not an eigenfunction of
; instead - being a polynomial of degree , can be expressed as
a linear combination of (in fact is
even/odd for even/odd k's): with we obtain
and hence
Rearranging we get (here, 
indicates halving the last term)
and similarly for the second derivative
The amount of work to carry out the differentiation in this form is
operations which destroys the efficiency. Instead, we can
employ the recursion relation which follows directly from (app_cheb.44)
To see this in a different way we note that
which leads to
and hence
as asserted. In general we have
With this, can be evaluated using operations, and the
differentiated polynomial at the grid points is computed using another cos-FFT
employing operations
with spectral/exponential error
The matrix representation of Chebyshev differentiation, , takes the almost
antisymmetric form (here except for )