Advanced Analysis, Notes 3: Hilbert spaces (application: Fourier series)
by Orr Shalit
Consider the cube . Let be a function defined on . For every , the th Fourier coefficient of is defined to be
where for and we denote . The sum
is called the Fourier series of . The basic problem in Fourier analysis is whether one can reconstruct from its Fourier coefficients, and in particular, under what conditions, and in what sense, does the Fourier series of converge to .
One week into the course, we are ready to start applying the structure theory of Hilbert spaces that we developed in the previous two lectures, together with the Stone-Weierstrass Theorem we proved in the introduction, to obtain easily some results in Fourier series.
1. An approximation result
Recall that we defined to be the completion of the inner product space with respect to the inner product
Recall also that this ends up being the same space as one encounters in a course in measure theory. The reader may choose either definition: what we will require in this lecture is only two facts. First, that is dense in , and second, that is complete (so it is a Hilbert space).
A simple computation shows that the collection of functions is an orthonormal system in . We clearly have , i.e., the Fourier coefficients of are what we defined last lecture to be the (generalized) Fourier coefficients of with respect to the system .
We let denote set of all complex trigonometric polynomials, that is, all the finite sums of the form
We also let denote the set of continuous periodic functions on , that is, the functions for which , , etc., for all . The spaces and are contained in and therefore also in . We denote by the sup norm in and by the Hilbert space norm in .
Lemma 1: is dense in in the norm.
Proof: This follows immediately from the trigonometric approximation formula from the introduction, together with the identities and . Alternatively, one may apply the complex version of the Stone-Weierstrass Theorem to the closure of .
Corollary 2: is dense in in the norm.
Proof: Clearly, for every we have , and the result follows.
Lemma 3: is dense in in the norm.
Proof: Let , and denote . Let be a function that satisfies:
Such a function is easy to construct explicitly: for example , where . If then and
and the right hand side is less than , which can be made as small as you wish.
Corollary 4: is dense in .
Proof: Let and be given. Let that approximates to within , let that approximates to within , and let that approximates to within . Then
2. Convergence of Fourier series
Theorem 5 ( convergence of Fourier series): For any , the Fourier series of converges to in the norm. In particular, is a complete orthonormal system and the following hold:
Remark: We use the notation .
Proof: The situation is similar to the one in linear algebra: the theory is so neat and tight that one can give several slightly different quick proofs.
First proof: By Corollary 4, (3) of Proposition 19 in Notes 2 holds. Therefore the equivalent (1) and (2) of Proposition 19 hold, which correspond to (I) and (II) here. Completeness is immediate from either (I) or (II).
Second proof: By Corollary 4, the system is complete. Indeed, assume that . The . The corollary implies that there is a sequence in . Thus
whence . Now Theorem 21 of Notes 2 implies the result.
Theorem 5, although interesting, elegant and useful, leaves a lot of questions unanswered. For example, what about pointwise convergence? For functions, only almost everywhere convergence makes sense, and it is a fact (Carleson’s Theorem) that the Fourier series of every converges almost everywhere to . Carleson’s Theorem requires far more delicate analysis then the norm convergence result that we obtained. Another natural question is what about uniform convergence? It turns out that Theorem 5 is powerful enough to imply the following beautiful result.
Theorem 6: For every , the Fourier series of converges uniformly to .
Proof: This follows from Theorem 5 and some rather basic first year analysis. It is left for the student as Exercise A, so you get to feel the power we have accumulated in your own hands.
Remark: Note that if a Fourier series converges uniformly, then the limit must be in . On the other hand, we will see in a later lecture that there are functions in whose Fourier series diverges at a dense set of points in . Thus, Theorem 6 is a pretty good theorem.