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Spectral accuracy

Define the Sobolev space consisting of tex2html_wrap_inline11001-periodic functions for which their first s-derivatives are tex2html_wrap_inline11009-integrable; set the corresponding -inner product as
The essential ingredient here is that the system - which was already shown to be complete in , is also a complete system in for any tex2html_wrap_inline11039. For orthogonality we have
The Fourier expansion now reads
where the Fourier coefficients, , are given by
We integrate by parts and use periodicity to obtain

and together with (app_fourier.20) we recover the usual Fourier expansion we had before, namely
The completion of in gives us the Parseval's equality (compare (app_fourier.15)) which in turn implies
we conclude from (app_fourier.24), that for any we have
Note that . This kind of estimate is usually referred to by saying that the Fourier expansion has spectral accuracy:

-- the error tends to zero faster than any fixed power of N, and is restricted only by the global smoothness of w(x).

We note that as before, this kind of behavior is linked directly to the spectral decay of the Fourier coefficients. Indeed, by Cauchy-Schwartz inequality
In fact more is true. By Parseval's equality

and hence by the Riemann-Lebesgue lemma, the product is not only bounded (as asserted in (app_fourier.27), but in fact it tends to zero,

Thus, tends to zero faster than for all . This yields spectral convergence, for

i.e., we get slightly less than (app_fourier.26),

Moreover, there is a rapid convergence for derivatives as well. Indeed, if then for we have

with Thus, for each derivative we ``lose'' one order in the convergence rate.

As a corollary we also get uniform convergence of for -functions w(x), with the help of Sobolev-type estimate
(Proof: Write with , and use Cauchy-Schwartz to upper bound the two integrals on the right.)

Utilizing (app_fourier.29) with we find

In particular, we conclude that for any we have, (in fact, s > 1/2 will do - consult (2.5.22) below)

In closing this section, we note that the spectral-Fourier projection, , can be rewritten in the form

Thus, the spectral projection is given by a convolution with the so-called Dirichlet kernel,
Now (app_fourier.30) reads

next up previous contents
Next: The Periodic Problem -- Up: The Periodic Problem -- Previous: The Periodic Problem --

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