Ptolemaics meetings 4 & 5 & 6 ; pt I

These last ones have been quite interesting meetings, I’m happy about how the whole thing is turning out. Sadly I’m very slow at typing and working out the ideas, so I have to include three different meetings in one. Since the notes are getting incredibly long, I’ll have to split it in at least two parts.I include the pdf version of it, in case it makes it any easier to read.

ptolemaics meeting 4 & 5 & 6 pt I

Let me get finally into the time frequency of the Walsh phase plane. I won’t include many proofs as they are already well written in Hytönen’s notes (see previous post). My main interest here is the heuristic interpretation of them (disclaimer: you might think I’m bullshitting you at a certain point, but I’m probably not). Ideally, it would be very good to be able to track back the train of thoughts that went in Fefferman’s and Thiele-Lacey’s proofs.

Sorry if the pictures are shit, I haven’t learned how to draw them properly using latex yet.

1. Brush up

Recall we have Walsh series for functions {f \in L^2(0,1)} defined by

\displaystyle W_N f(x) = \sum_{n=0}^{N}{\left\langle f,w_n\right\rangle w_n(x)},

the (Walsh-)Carleson operator here is thus

\displaystyle \mathcal{C}f(x) = \sup_{N\in \mathbb{N}}{|W_N f(x)|},

and in order to prove {W_N f(x) \rightarrow f(x)} a.e. for {N\rightarrow +\infty} one can prove that

\displaystyle \|\mathcal{C}f\|_{L^{2,\infty}(0,1)} \lesssim \|f\|_{L^2(0,1)}.

There’s a general remark that should be done at this point: the last inequality is equivalent to

\displaystyle \left|\left\langle\mathcal{C}f, \chi_E\right\rangle\right| = \left|\int_{E}{\mathcal{C}f}\,dx\right| \lesssim |E|^{1/2}\|f\|_{L^2(0,1)}

to hold on every measurable {E} (of finite measure).
Continue reading

Ptolemaics meetings 2 & 3

I’m very far behind in this, I’ve got to admit I’m busier than last year and haven’t coped with it completely yet. Luckily we’re progressing slowly in order to understand better and that gives me the opportunity to merge together the blog posts for meetings 2 & 3.

As I said in the previous post, the goal for now is to understand the proof of {L^2\rightarrow L^{2,\infty}} boundedness of Carleson’s operator (through time freq. analysis). As an introduction to the real thing we’ve started from the simpler case of the Walsh transform, or Walsh series, or Walsh phase plane, whatever you want to call it. It’s easier because all the nasty technicalities disappear but the ideas needed are already in there, that’s why it propaedeutic. We’re following Hytönen’s notes as suggested by Tuomas (you can find them here: http://wiki.helsinki.fi/pages/viewpage.action?pageId=79564963). An alternative is Tao’s lecture notes (lecture 5 in particular) for course Math254A W’ 01 (http://www.math.ucla.edu/~tao/254a.1.01w/) which are quite nice – as all of his stuff. The main differences are in that Hytönen proves every single statement, and he deals with the Walsh series (analogue of the Fourier series) while Tao deals with the Walsh transform (analogue of the Fourier transform). Also, Hytönen then goes on to prove the full euclidean case, while Tao doesn’t.

The Walsh operators are best described as operators on the real line with a different field structure. One works on {\mathbb{Z}_2[[X]]}, i.e. the Laurent series with coefficient in {\mathbb{Z}_2}, which can be identified with the (binary expression of) positive reals by

\displaystyle a_N \cdots a_0 . a_{-1} a_{-2} \cdots \equiv a_N X^N + \ldots a_1 X + a_0 + \frac{a_{-1}}{X} + \frac{a_{-2}}{X^2} + \ldots .

Continue reading

Ptolemaics meeting #1

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

\displaystyle \sum_{n=0}^{N}{\left(a_n \cos{2\pi n x} + b_n \sin{2\pi n x}\right)},

or, with a better notation,

\displaystyle \sum_{n=-N}^{N}{c_ n e^{2\pi i n x}}.

Continue reading