Advanced Analysis, Notes 3: Hilbert spaces (application: Fourier series)

Consider the cube $K := [0,1]^k \subset \mathbb{R}^k$. Let $f$ be a function defined on $K$.  For every $n \in \mathbb{Z}^k$, the $n$th Fourier coefficient of $f$ is defined to be

$\hat{f}(n) = \int_{K} f(x) e^{-2 \pi i n \cdot x} dx ,$

where for $n = (n_1, \ldots, n_k)$ and $x = (x_1, \ldots, x_k) \in K$ we denote $n \cdot x = n_1 x_1 + \ldots n_k x_k$.  The sum

$\sum_{n \in \mathbb{Z}^k} \hat{f}(n) e^{2 \pi i n \cdot x}$

is called the Fourier series of $f$. The basic problem in Fourier analysis is whether one can reconstruct $f$ from its Fourier coefficients, and in particular, under what conditions, and in what sense, does the Fourier series of $f$ converge to $f$.

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 $L^2(K)$ to be the completion of the inner product space $C(K)$ with respect to the inner product

$(f,g) = \int_K f(x) \overline{g(x)} dx .$

Recall also that this ends up being the same space $L^2(K)$ 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 $C(K)$ is dense in $L^2(K)$, and second, that $L^2(K)$ is complete (so it is a Hilbert space).

A simple computation shows that the collection of functions $\{e^{2 \pi i n \cdot x}\}_{n \in \mathbb{Z}^k}$ is an orthonormal system in $L^2(K)$. We clearly have $\hat{f}(n) = (f, e^{2 \pi i n \cdot x})$, i.e., the Fourier coefficients of $f$ are what we defined last lecture to be the (generalized) Fourier coefficients of $f$ with respect to the system  $\{e^{2 \pi i n \cdot x}\}_{n \in \mathbb{Z}^k}$.

We let $P$ denote set of all complex trigonometric polynomials, that is, all the finite sums of the form

$\sum_n a_n e^{2 \pi i n \cdot x}$.

We also let $C_{per}(K)$ denote the set of continuous periodic functions on $K$, that is, the functions $f$ for which $f(0, x_2, \ldots, x_k) = f(1, x_2, \ldots, x_k)$, $f(x_1, 0, x_3, \ldots, x_k) = f(x_1, 1, x_3, \ldots, x_k)$, etc., for all $x = (x_1. \ldots, x_k) \in K$ . The spaces $P$ and $C_{per}(K)$ are contained in $C(K)$ and therefore also in $L^2(K)$. We denote by $\| \cdot \|_\infty$ the sup norm in $C(K)$ and by $\| \cdot \|_2$ the Hilbert space norm in $L^2(K)$.

Lemma 1: $P$ is dense in $C_{per}(K)$ in the $\| \cdot \|_\infty$ norm.

Proof: This follows immediately from the trigonometric approximation formula from the introduction, together with the identities $2 i \sin t = e^{it} - e^{-it}$ and $2 \cos t = e^{it} + e^{-it}$. Alternatively, one may apply the complex version of the Stone-Weierstrass Theorem to the closure of $P$.

Corollary 2: $P$ is dense in $C_{per}(K)$ in the $\| \cdot \|_2$ norm.

Proof: Clearly, for every $f \in C(K)$ we have $\|f\|_2 \leq \|f\|_\infty$, and the result follows.

Lemma 3: $C_{per}(K)$ is dense in $C(K)$ in the $\| \cdot \|_2$ norm.

Proof: Let $\epsilon > 0$, and denote $L = [\epsilon, 1-\epsilon]^k$. Let $g \in C(K)$ be a function that satisfies:

1. $0 \leq g \leq 1$,
2. $g\big|_L = 1$,
3. $g\big|_{\partial K} = 0$.

Such a function is easy to construct explicitly: for example $g(x) = \epsilon^{-1} \left( d(x,\partial K) \wedge \epsilon\right)$, where $d(x, \partial K) = \inf \{\|x - y\| : y \in \partial K\}$. If $f \in C(K)$ then $fg \in C_{per}(K)$ and

$\|f - fg\|_2^2 = \int_K |f|^2 |1-g|^2 dx \leq \|f\|_\infty^2 \int_{K \setminus L} 1 dx$

and the right hand side is less than $\|f\|_\infty^2 (1 - (1-2 \epsilon)^k)$, which can be made as small as you wish.

Corollary 4: $P$ is dense in $L^2(K)$.

Proof: Let $f \in L^2(K)$ and $\epsilon > 0$ be given. Let $g \in C(K)$ that approximates $f$ to within $\epsilon/3$, let $h \in C_{per}(K)$ that approximates $g$ to within $\epsilon/3$, and let $p \in P$ that approximates $h$ to within $\epsilon/3$. Then

$\|f - p\|_2 \leq \|f - g\|_2 + \|g - h\|_2 + \|h - p\|_2 < \epsilon .$

2. Convergence of Fourier series

Theorem 5 ($L^2$ convergence of Fourier series): For any $f \in L^2(K)$, the Fourier series of $f$ converges to $f$ in the $\|\cdot \|_2$ norm. In particular, $\{e^{2 \pi i n \cdot x}\}_{n \in \mathbb{Z}^k}$ is a complete orthonormal system and the following hold:

(I)      $\|f\|_2^2 = \sum |\hat{f}(n)|^2$

(II)      $\lim_{N\rightarrow \infty} \|f(x) - \sum_{|n| \leq N} \hat{f}(n) e^{2 \pi i n \cdot x}\| = 0 .$

Remark: We use the notation $|N| = |n_1| + \ldots + |n_k|$.

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 $\{e^{2 \pi i n \cdot x}\}_{n \in \mathbb{Z}^k}$ is complete. Indeed, assume that $f \perp \{e^{2 \pi i n \cdot x}\}_{n \in \mathbb{Z}^k}$. The $f \perp P$.  The corollary implies that there is a sequence in $P \ni p_n \rightarrow f$. Thus

$\langle f, f \rangle = \lim_n \langle f, p_n \rangle = 0 ,$

whence $f = 0$. 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 $L^2$ functions, only almost everywhere convergence makes sense, and it is a fact (Carleson’s Theorem) that the Fourier series of every $f \in L^2$ converges almost everywhere to $f$. 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 $f \in C_{per}(K) \cap C^1(K)$, the Fourier series of $f$ converges uniformly to $f$.

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 $C_{per}(K)$. On the other hand, we will see in a later lecture that there are functions in $C_{per}([0,1])$ whose Fourier series diverges at a dense set of points in $[0,1]$. Thus, Theorem 6 is a pretty good theorem.