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The Periodic Problem -- The Fourier Expansion


Consider the first order Sturm-Liouville (SL) problem
augmented with periodic boundary conditions
It has an infinite sequence of eigenvalues, tex2html_wrap_inline11133, with the corresponding eigenfunctions tex2html_wrap_inline11135. Thus, tex2html_wrap_inline11137 are the eigenpairs of the differentiation operator tex2html_wrap_inline11139 in tex2html_wrap_inline11141, and they form a in this space -- completeness in the sense described below.

Let the space tex2html_wrap_inline11141 be endowed with the usual Euclidean inner product
Note that tex2html_wrap_inline11135 are orthogonal with respect to this inner product, for
Let tex2html_wrap_inline11147 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 tex2html_wrap_inline11151-the space of trigonometric polynomials of degree tex2html_wrap_inline11153: gif
here tex2html_wrap_inline11161 and tex2html_wrap_inline11163 are the usual Fourier coefficients given by
Since tex2html_wrap_inline11165 is orthogonal to the tex2html_wrap_inline11151-space:
it follows that for any tex2html_wrap_inline11169 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 tex2html_wrap_inline11147 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 tex2html_wrap_inline11009 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 tex2html_wrap_inline11009-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 tex2html_wrap_inline11009-functions. Yet, if we agree to assume sufficient smoothness, we find the convergence of spectral-Fourier projection to be very rapid, both in the tex2html_wrap_inline11009 and the pointwise sense. To this we proceed as follows.

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Eitan Tadmor
Thu Jan 22 19:07:34 PST 1998