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The (Pseudo)Spectral Fourier Expansions - Exponential Accuracy


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 tex2html_wrap_inline11091 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 tex2html_wrap_inline11001-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

Eitan Tadmor
Thu Jan 22 19:07:34 PST 1998