Define the Sobolev space consisting of -periodic
functions for which their first *s*-derivatives are -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 . 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

Since

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, __

__
Moreover, there is a rapid convergence for derivatives as well. Indeed, if
then for we have
Hence
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
where
Thus, the spectral projection is given by a convolution with the so-called
__

Thu Jan 22 19:07:34 PST 1998