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, 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
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
Dirichlet kernel,
Now (app_fourier.30) reads