The n-Category Café

Re: Carleson’s Theorem

I don’t think there is much simplification to the proof of Carleson’s theorem in restricting attention to continuous functions, even if one only wants convergence somewhere rather than almost everywhere. It’s now realised that pointwise convergence questions of $S_n f(x)$ are closely related to the boundedness properties of the Carleson maximal operator $$S^* f(x) := \sup_n |S_n f(x)|.$$ (This is only a sublinear operator rather than a linear operator, but it’s still an operator nonetheless.) Roughly speaking, if one can prove a non-trivial bound on $S^* f$ (and in particular keep it finite almost everywhere) for all $f$ in a function space (e.g. $L^p$), then it is a relatively routine matter to demonstrate almost everywhere convergence in $L^p$; and conversely, if no such bound exists, then it is likely that (with perhaps a bit of nontrivial trickery) one can eventually cook up a counterexample in this function space for which one has pointwise convergence nowhere. If one has a bit of regularity, e.g. $C^1$, then bounding $S^* f$ is relatively straightforward, but in spaces with zero regularity (E.g. $L^p$ or $C^0$) it’s much more difficult. The key problem here is the modulation invariance of $S^*$: if one multiplies $f$ by a character $e^{2\pi i kx}$, this essentially does not change $S^*$. (This invariance is more apparent if one replace $S_n$ with to the closely related double sum $S_{m,n} f(x) = \sum_{-m}^n \hat f(k) e^{2\pi i kx}$ in the definition of $S^*$.) The spaces $L^p$ and $C^0$ are also modulation invariant, and this basically forces any proof of almost everywhere (or even somewhere) pointwise convergence in these spaces to also be modulation invariant, which rules out a lot of standard techniques and requires instead tools such as time-frequency analysis.

Re: Carleson’s Theorem

It at least is within reach of an undergraduate class to prove more general results about everywhere convergence than Dirichlet’s theorem, for example for Hölder continuous functions.

Re: Carleson’s Theorem

So will the café commenting interface also remember names, emails and urls like it never did? (Not to get too off topic; we could discuss at the n-forum instead.)

Re: Carleson’s Theorem

I think the lecture note proof of the decay rate for the Fourier coefficients of Lipschitz functions is needlessly complicated. If we consider, more generally, a Holder continuous function $f$ of order $\alpha$ on $\mathbb{T}$ then then decay rate $|\hat{f}(k)| \lesssim (1+|k|)^{-\alpha}$ is an obvious consequence of the identity $$\hat{f}(k) = -\int_{\mathbb{T}} f(x - \tfrac{1}{2k})e^{-2\pi i kx} \,dx$$ However, I don’t think the summation method argument adapts if you assume the coefficients are just $O(k^{-\alpha})$.

Re: Carleson’s Theorem

With regards to Carleson’s theorem a MUCH simpler result along similar lines is the almost everywhere convergence of the Fejer means of an $L^1(\mathbb{T})$ function. The reason I mention this is that proof has the same general outline as that of Carleson’s Theorem - here we consider the Fejer maximal function, defined in an analogous way to the Carleson maximal function, and try to show it satisfies some interesting bound. I mention this as a point of interest rather than something suitable for inclusion in the course.

Re: Carleson’s Theorem

Even if the usual Fourier series of a continuous function $f: \mathbf{T} \to \mathbb{C}$ does not always converge pointwise to $f$ (much less uniformly to $f$), it might be nice to point out to your students that the averages of the finite Fourier approximations

$$A_N(f) = \frac1{N+1} \sum_{n=0}^N S_n(f)$$

do converge to $f$ uniformly! These are the Fejer means that Jonathan referred to.

The way I think of this goes as follows. (All of this is really well known. Certainly it’s much better known to the analysts reading this thread than it is to a semi-ignorant category theorist like myself. And for all I know it’s covered in Körner’s book – I’ve never looked at the book, but have heard very nice things about it.)

First, there’s a Banach algebra $L^1(\mathbf{T})$ where on the unit circle $\mathbf{T}$ we use the normalized Haar measure, and the multiplication on $L^1$ is given by convolution. (This algebra doesn’t have an identity, but we can adjoin one, and it is useful to think of it as the Dirac distribution supported at the identity $1 \in \mathbf{T}$.) The spaces $L^p(\mathbf{T})$, for $1 \leq p \leq \infty$ are Banach modules under convolution product: the convolution product

$$\ast: L^1(\mathbf{T}) \times L^p(\mathbf{T}) \to L^p(\mathbf{T})$$

is bilinear and continuous because we have ${\|f \ast g\|}_p \leq {\|f\|}_1 {\|g\|}_p$.

Here are some useful examples of convolution:

• Let $e_n: \mathbf{T} \to \mathbb{C}$ denote the character $z \mapsto z^n$, or if you like, the character $x \mapsto e^{i n x}$ where we identify $\mathbf{T}$ with $\mathbb{R}/2\pi \mathbb{Z}$. Let $f: \mathbf{T} \to \mathbb{C}$ be continuous, say. Then

  $$(e_n \ast f)(y) = \frac1{2\pi} \int_{-\pi}^{\pi} e^{i n (y-x)} f(x) d x = (\frac1{2\pi} \int_{-\pi}^{\pi} e^{-i n x} f(x) d x)\; e^{i n y}$$

  where the integral in the parentheses gives the $n^{th}$ Fourier coefficient. Thinking of $e_n$ and $f$ as living in $L^2(\mathbf{T})$, this Fourier coefficient is the Hilbert space pairing $\langle e_n, f \rangle$, and so we get the neat little formula

  $$e_n \ast f = \langle e_n, f \rangle e_n.$$

• The operator $S_n: f \mapsto S_n(f)$ is defined by $$f \mapsto \sum_{k = -n}^n \langle e_k, f \rangle e_k$$ where we orthogonally project $f$ onto the subspace of $L^2(\mathbf{T})$ spanned by the orthonormal elements $e_{-n}, e_{-n+1}, \ldots, e_n$. Using the example above, we see that the operator $S_n$ can be described as the result of convolving with the function

  $$D_n = e_{-n} + e_{-n+1} + \ldots + e_n,$$

  i.e.,

  $$S_n(f) = D_n \ast f.$$

  This $D_n$ is called the $n^{th}$ Dirichlet kernel. It’s manifestly a finite geometric series, and thus we easily compute

  $$D_n(x) = e^{-i n x} \frac{e^{(2 n + 1)i x} - 1}{e^{i x} - 1} = \frac{e^{(n + 1/2) i x} - e^{-(n + 1/2)i x}}{e^{(1/2)i x} - e^{-(1/2)i x}} = \frac{\sin((n + 1/2)x)}{\sin((1/2)x)}.$$

  This (a) spikes to the value $2n + 1$ at $x = 0$, and (b) has “mass”

  $$\frac1{2\pi}\int_{-\pi}^{\pi} e_{-n}(x) + \ldots + e_0(x) + \ldots + e_n(x) d x = \frac1{2\pi} \int_{-\pi}^{\pi} e_0(x) d x = 1,$$

  and these two facts (a), (b) mean $D_n$ behaves something like an approximate Dirac distribution supported at $0$ – except it doesn’t taper off to zero a small distance away from $0$.

Now for a brief interlude on this business of “Dirac distribution” as identity. To repeat: the Banach algebra $L^1(\mathbf{T})$ has no identity for the convolution product (which is what the Dirac distribution $\delta$ would be, except that this “mystical” $\delta(x)$, which is infinite at $x=0$, and zero away from $x=0$, and has integral $1$, is not represented by an $L^1$ function). But, as a workaround, we can speak of approximate identities. A sequence of $L^1$ functions $K_n$ is an approximate identity if

1. For all $n$ we have $$\frac1{2\pi} \int_{-\pi}^{\pi} K_n(x) d x = 1;$$
2. For all $\epsilon, \delta \gt 0$ there exists $N$ such that $$\frac1{2\pi} \int_{[-\pi, \pi]\backslash [-\epsilon, \epsilon]} {|K_n(x)|}\;\; d x \lt \delta,$$ whenever $n \geq N$.

The second condition says that away from the identity element $0$, an approximate identity is close to zero (in the $L^1$ sense), while still having total mass $1$ by the first condition.

Lemma: Let $\{K_n\}$ be an approximate identity. For each of the Banach modules $L^p(\mathbf{T})$ over $L^1(\mathbf{T})$, $1 \leq p \leq \infty$, we have

$$\underset{n \to \infty}{\lim} K_n \ast f = f$$

for all $f \in L^p(\mathbf{T})$. (This justifies the term “approximate identity”.)

It should also be said that convolutions $K \ast f$ inherit the regularity behavior of $K$ or $f$. For example, if $K$ or $f$ is of class $C^n$, then so is $K \ast f$ (even if $f$ or $K$, respectively, isn’t). Consequently, if $f$ is continuous, then convergence $K_n \ast f \to f$ in the $L^\infty$ norm becomes uniform convergence of continuous functions.

Now we head to our third example.

• Define an operator (the $N^{th}$ Cesaro or Fejer mean) on continuous functions $f: \mathbf{T} \to \mathbb{C}$ by averaging the $S_n(f)$:

  $$K_N(f) = \frac1{N+1} \sum_{n=0}^N S_n(f).$$

  Thus $K_N$ is the result of convolving with an average of Dirichlet kernels: $K_N(f) = F_N \ast f$ where

  $$F_N = \frac1{N+1} \sum_{n=0}^N D_n.$$

  This $F_N$ is called a Fejer kernel. Now here is a pretty neat fact:

  $$F_N = \frac1{N+1} D_N^2.$$

  Indeed, $D_N^2 = \sum_{n=0}^N D_n$ for essentially the same reason that $11111^2 = 123454321$ – the proof of either can be left as an exercise. Hence we have

  $$F_N = \frac1{N+1} \frac{\sin^2((N+1/2) x)}{\sin^2((1/2)x)}$$

  and the positive functions $\{F_n\}$ form an approximate identity. For, each of the $F_n$ has mass $1$ since each is an average of Dirichlet kernels which have mass $1$, and given any $\delta, \epsilon \gt 0$ there exists $N$ such that

  $${|F_n(x)|} \leq \frac1{(n+1)\sin^2(\epsilon/2)} \leq \delta$$

  for all $x \in [-\pi, \pi] \backslash [-\epsilon, \epsilon]$ and $n \geq N$.

  Therefore, by the lemma above, for $f: \mathbf{T} \to \mathbb{C}$ continuous, the Cesaro means $K_n(f)$ converge uniformly to $f$.

Re: Carleson’s Theorem

Tom, one other thing. Did you ever read the review of Körner’s book by Gavin Brown? It’s absolutely delightful.

Re: Carleson’s Theorem

Regarding the “false beliefs” about pointwise convergence mentioned in the post: the false beliefs of Fourier form part of the backdrop of the second story that Lakatos tells in his Proofs and Refutations (his Appendix 1: Cauchy on the ‘Principle of Continuity’).