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The Periodic Problem -- The Fourier Expansion

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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.

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 or equivalently, 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.    Next: Spectral accuracy Up: SPECTRAL APPROXIMATIONS Previous: SPECTRAL APPROXIMATIONS