Together with some other PG students in the Harmonic Analysis working group, we’ve decided (it was Kevin’s idea originally) to set up a weekly meeting to learn about topics of harmonic analysis we don’t get to see otherwise (it works quite well as an excuse to drink beer, too). The topic we settled on arose pretty much by itself: it turned out that basically everybody was interested in time-frequency analysis on his own, either through Carleson’s theorem or some other related stuff. So we decided to learn about time-frequency analysis.
Last tuesday we had our first meeting: it was mainly aimed at discussing the arrangements to be made and what to read before next meeting, but we sketched some motivational introduction (it was quite improvised, I’m afraid); see below. Also, it was Odysseas that came up with the name. I think it’s quite brilliant: Ptolemy was the first to introduce the systematic use of epicycles in astronomy, and – as the science historian Giovanni Schiapparelli noticed – epicycles were nothing but the first historical appearance of Fourier series. That’s why they offered such accurate predictions even though the theory was wrong: by adding a suitable number of terms you can describe orbits within any amount of precision. Thus, from Carleson’s result you can go all the way back to Ptolemy: therefore Ptolemaics. Odysseas further added that Ptolemy’s first name was Claudius, like the roman emperor that first began the effective conquest of Britain; but that’s another story.
I will incorporate below a post I was writing for this blog about convergence of Fourier series, so it will be quite long in the end. Sorry about that, next posts will probably be way shorter.
1. Fourier series trivia
First some trivia of Fourier series as to brush up.
One wishes to consider approximations of functions (periodic of period 1) by means of trygonometric polynomials
or, with a better notation,
If one restricts himself to functions for example (where is the unit circle, or if you prefer), he’s immediately led to consider the projections onto the spaces (the vectors are easily verified to be orthonormal to each other),
and it can be noted that
where is called the Dirichlet kernel and has expression
The study of projections is thus reduced to that of a sequence of convolution operators. Notice how , so one can think of it as a particular approximation of unity. As it is usual, one is then concerned with convergence issues: does converge to for ? and in what sense? Well, the most ubiquitous notions are those of pointwise convergence (or a.e. convergence) and convergence. The first one is harder to establish and is the content of the famous theorem by Carleson (see also later posts). We concentrate on this convergence for the moment.
First of all, can be computed explicitely as a geometrical series:
Now, it can be proved under some regularity hypothesis on that
(so that if is also continuous in a point, one has convergence in that point). To understand the regularity hypothesis let’s see where they arise from. First, change by periodicity the interval of integration to the symmetric , for the sake of convenience. One can exploit the fact that by writing
and since one has the same identity with in place of , so that
We know that the behaviour of (in the Dirichlet kernel) is essentially when is close to , so in order to make that more explicit write
We don’t really worry about the factor as its absolute value is of size in . So, the r.h.s. will tend to zero (and thus we’ll have ) by the Riemann-Lebesgue lemma if
is in . This is what is called a Dini condition – automatically satisfied in the (strong) assumption that be Lipschitz continuous. Therefore, a function satisfying Dini condition in will have convergent Fourier series in there,
Notice that without exploiting the symmetry of one could ask just for the stronger condition
which is sometimes referred to as Dini condition as well. It should also be noted that it is enough to have that the function above is in for some , not necessarily on the whole interval therefore.
A different criterion (the one for which we have convergence to the average of left and right limits) is that be of bounded variation in a nbhd. of . This is called Jordan’s criterion. Its proof relies on the fact that function of bounded variation on the real line can be written as the difference of two monotonic functions.
2. Convergence failures
The above digression settles the argument for functions with enough regularity. On the other hand, it can be proved that there are continuous functions (not just piecewise like in Jordan’s criterion) such that the Fourier series diverges at a chosen point . It can be proved by the contrapositive of Banach-Steinhaus theorem: if the operator norms of the partial series are unbounded, then there exists at least one function for which . This is because is a Banach space with norm , so the hypothesis of Banach-Steinhaus are satisfied. It suffices to calculate the operator norms of . Assume , then
and it’s not that hard to build a continuous function with that shows that the norm of is actually . This norm grows logaritmically in , so we’re done. That the growth is about logaritmical in can be guessed by the following: by rearranging the terms one can see that
so that with some manipulation
this last espression can be bounded by plus a constant, as by symmetry
and the function is easily seen to be bounded independently of on . Thus
Of course one could compute it directly.
Now the continuous function of which we just proved existence has Fourier series divergent in 0. How bad can it get for continuous functions? Well, as by Carleson’s theorem, functions in (and thus also in ) have Fourier series convergent almost everywhere, so that the worst that can happen is that the set of divergence is of measure zero.
Is it better than that or is this the best we can say? the case turns out to be the second. Odysseas directed me to the book An introduction to Harmonic Analysis by Yitzhak Katznelson (first edition in 1968). It is proved in there that a sufficiently regular space of functions either contains a function s.t. diverges everywhere (see below) or the the Fourier series of all functions in converge (only) a.e.. Therefore it makes sense to ask for a.e. convergence only and not pointwise (which is trivial for spaces because the functions are actually just equivalence classes modulus zero measure sets, but not just as trivial in general – for for example). More rigorously: a set of divergence for is a set such that there exists the Fourier series of which diverge for all . Equivalently,
Then the theorem goes
Theorem 1 Let be a homogeneous Banach space  of functions on such that . Then either is a set of divergence for or the sets of divergence of are precisely the subsets of measure zero.
Informally, if is a set of divergence of positive measure, every translate is again a set of divergence and then by taking the union over all rational translates it can be proved we still obtain a set of divergence, and this set has measure . Then one proves separately that every zero measure set is a divergence set for and we’re done. I might have to say more about this next week, we’ll see.
Anyhow, as a quick survey, the result of Carleson (and Hunt’s extension to for ) implies that the Fourier series of functions in converge a.e., so that we are in the second situation outlined by the theorem. We find ourselves in the first one for , as Kolmogorov showed with his famous example of a function in such that is everywhere divergent. Details on the construction can be found in Katznelson’s book. In the meeting we asked ourselves if there’s anything deep under Kolmogorov’s example but the shared opinion is that there isn’t – it’s just technical (although that’s open to be questioned – nobody remembered the construction at the time).
As a further note, it is still an open problem to determine the “critical” -space for a.e. convergence: as just said it’s in between and any other for . It has been proved that a.e. convergence holds for functions in , and on the other side there’s an example of a function in with everywhere divergent Fourier series. It is conjectured that the largest space in which a.e. convergence holds is .
3. Carleson’s theorem
Personally, I got interested in Carleson’s theorem because the proof Lacey and Thiele gave is a strong example of time-frequency analysis. Carleson’s original proof, as I understand it, relied on a subtle and quite complicated decomposition of ; historically, a second proof was given by Fefferman considering linearization of the Carleson operator instead. This proof relied heavily on time-frequency analysis, which was made explicit in Lacey and Thiele’s proof.
One defines Carleson’s maximal operator (which already appeard above) as
If one can prove the weak type estimate
then convergence a.e. follows by a standard density argument: namely, since convergence holds for functions and they are dense in , take s.t. and define function
Then since one has
and by arbitrariness of it follows a.e..
As seen above, we can “factor out” as it is of roughly constant size in , and work with the equivalent kernel
Now, write for and notice that
thus by Fubini
If one were to replace by this would be nothing but Fourier inversion formula (interpret as the Fourier transform on of the function ). Hence we’ll concentrate on this, and in particular we can obviously content ourselves with the one-sided Carleson operator (which we denote by the same letter from now on)
We state Carleson’s theorem once again (but this is the last one, I promise)
Theorem 2 (Carleson, 1966) Carleson’s maximal operator is bounded. As a corollary, the Fourier inversion formula holds pointwise a.e. for functions in and the Fourier series of functions in converge a.e..
The goal of the student group is to understand the proof of this theorem in depth. We shall start from the mock-case of the Walsh plane next week, where all the ideas are already present. But meanwhile, as a motivating digression, prof. Wright showed us how a proof of the maximal Hausdorff-Young inequality can be given in just a few lines thanks to a (tricky) idea of Christ and Kiselev (see ); this is the subject of next (and last) section.
4. Maximal Hausdorff-Young inequality
Instead of considering the inversion problem for the Fourier transform, consider the maximal version of it, namely
If one could prove this is of weak type , that would be equivalent to Carleson’s theorem, since (and we have Plancherel’s inequality). Thus Carleson’s theorem can be seen as the endpoint of this maximal Fourier operator. And for the range ? That would be the maximal version of Hausdorff-Young inequality. It turns out it holds, i.e. maps to boundedly for . What's extremely nice about this is that no complex interpolation between the endpoints is needed (one being Carleson's theorem) – it can be proved on its own. The result is not particular of the Fourier transform and it can be proved in the following general setting: let be a linear operator on with kernel , i.e.
and assume is smoothing, that is
for . Then the maximal operator
is bounded from to as well. The proof is based on a very clever and surprising idea. Normalize , and partition the real line in the following dyadic way: first define , then choose such that and define and ; repeat the splitting in equal parts according to the mass on both the intervals , which are the intervals of level 1, and so on. In the end we get a dyadic frame such that
and we call the collection of intervals of level by . Notice there’s a natural relation amongst the intervals , that of being a brother: namely the interval is a brother to if . Moreover we can distinguish two brothers in left and right one. Call the left brothers in .
Now we linearize the supremum, i.e. we take the measurable function such that realizes the supremum (or at least a positive fixed fraction of it) in , and we change point of view: now is a fixed function and we have to prove the inequality
with constant independent of . We split the integration on in the following way
(notice the second sum from the left has at most one term really) from which it follows
Now because the inner sum has at most one term we can substitute it with the supremum in ,
and further replace the supremum with the average
where the ‘s don’t depend on anymore! Then we’re done, since
so that it is summable in , because . This concludes the proof, as the bound is independent of as we wished. Notice the smoothing assumption on is fundamental here.
Now, go back to the case of the Fourier transform. It corresponds to kernel , and it is is smoothing on for if . Thus the above theorem applies and one has the maximal Hausdorff-Young inequality
The theorem breaks down exactly at because for this exponent the Fourier transform is no more smoothing.
That’s all for today.
 : it means a Banach subspace of (the largest space on ) with a norm which is translation invariant and continuous w.r.t. to translations.
 : “Maximal functions associated to filtrations”, M. Christ and A. Kiselev, Jour. of Func. Anal., Vol. 179, Issue 2, 1 February 2001, Pages 409–425.