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### Spectral accuracy

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

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,