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 )

Thu Jan 22 19:07:34 PST 1998