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Consider the first order Sturm-Liouville (SL) problem

augmented with periodic boundary conditions

It has an infinite sequence of eigenvalues, , with the
corresponding eigenfunctions . Thus, are the *eigenpairs* of the
differentiation operator in , and
they form a __ in this space --
completeness in the sense described below.
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Let the space
be endowed with the usual Euclidean inner product
Note that are orthogonal with respect
to this inner product,
for
Let be associated with its __

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Figure 2.1: Least-squares approximation
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Hence, solves the least-squares problem
i.e., is the best least-squares approximation to w. Moreover,
(app_fourier.11) with yields
and by letting we arrive at
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: An immediate consequence of (app_fourier.14) is the
Riemann-Lebesgue lemma, asserting that
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The system is in the sense that for any we have

which in view of (app_fourier.13), is the same as

Thus completeness guarantee that the spectral projections 'fill in' the relevant space.

The last equality establishes the convergence of the spectral-Fourier projection, , to w(x), whose difference can be (upper-)bounded by the following

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:
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We observe that the RHS tends to zero as a tail of a converging sequence,
i.e.,
The last equality tells us that the convergence rate depends on how fast the
Fourier coefficients, , decay to zero, and we shall quantify this
in a more precise way below.
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. What about pointwise convergence?
The -convergence stated in (app_fourier.17)
yields pointwise a.e. convergence for
subsequences; one can show that in fact
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The ultimate result in this direction states that , (no subsequences) for all , though a.e. convergence may fail if is only -integrable.

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The question of pointwise a.e. convergence is an extremely intricate issue
for arbitrary -functions.
Yet, if we agree to assume sufficient smoothness, we find the
convergence of spectral-Fourier projection to be very rapid,
both in the
and the pointwise sense. To this we proceed as follows.
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Thu Jan 22 19:07:34 PST 1998