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.
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 spectral representation in this system, i.e., the Fourier expansion
The truncated Fourier expansion
denotes the spectral-Fourier projection of w(x) into -the space of trigonometric polynomials of degree :
here and are the usual Fourier coefficients given by
Since is orthogonal to the -space:
it follows that for any we have (see Figure 2.1 )
Figure 2.1: Least-squares approximation
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
: An immediate consequence of (app_fourier.14) is the
Riemann-Lebesgue lemma, asserting that
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
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.
. What about pointwise convergence?
The -convergence stated in (app_fourier.17)
yields pointwise a.e. convergence for
subsequences; one can show that in fact
The ultimate result in this direction states that , (no subsequences) for all , though a.e. convergence may fail if is only -integrable.
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.