0

We have seen that the spectral and the pseudospectral approximations
enjoy what we called ``spectral accuracy'' - that is, the convergence rate is
restricted solely by the global smoothness of the data. The statement about
``infinite'' order of accuracy for functions is an
*asymptotic* statement. Here we show that in the analytic case the
error decay rate is in fact *exponential*.

To this end, assume that

is -periodic analytic in the strip .
The error decay rate in both the spectral and pseudospectral cases is
determined by the decay rate of the Fourier coefficients .
Making the change of variables we have for

the power series expansion

By the periodic analyticity of *w*(*z*) in the strip is found to be single-valued analytic in the
corresponding annulus

whose Laurent expansion is given in (err_exp.3):

This yields exponential decay of the Fourier coefficients

We note that the inverse implication is also true; namely an exponential decay
like (err_exp.6) implies the analyticity of *w*(*z*). Inserting this into (app_fourier.17)
yields

and similarly for the pseudospectral approximation

Note that in either case the exponential factor depends on the distance of the
singularity (lack of analyticity) from the real line. For higher derivatives
we likewise obtain

We can do even better, by taking into account higher derivatives, e.g.,

so that with

we have

and hence

Thu Jan 22 19:07:34 PST 1998