DMat0101, Notes 3: The Fourier transform on L^1

1. Definition and main properties.

For {f\in L^1({\mathbb R}^n)}, the Fourier transform of {f} is the function

\displaystyle \mathcal{F}(f)(\xi)=\hat{f}(\xi)=\int_{{\mathbb R}^n}f(x)e^{-2\pi i x\cdot \xi}dx,\quad \xi\in{\mathbb R}^n.

Here {x\cdot y} denotes the inner product of {x=(x_1,\ldots,x_n)} and {y=(y_1,\ldots, y_n)}:

\displaystyle x\cdot y=\langle x,y\rangle=x_1y_1+\cdots x_n y_n.

Observe that this inner product in {{\mathbb R}^n} is compatible with the Euclidean norm since {x\cdot x=|x|^2}. It is easy to see that the integral above converges for every {\xi\in{\mathbb R}^n} and that the Fourier transform of an {L^1} function is a uniformly continuous function.

Theorem 1 Let {f,g\in L^1({\mathbb R}^n)}. We have the following properties.

(i) The Fourier transform is linear {\widehat{f+g}=\hat f + \hat g} and {\widehat{cf}=c \hat f} for any {c\in{\mathbb C}}.

(ii) The function {\hat f(\xi)} is uniformly continuous.

(iii) The operator {\mathcal F} is bounded operator from {L^1({\mathbb R}^n)} to {L^\infty({\mathbb R}^n)} and

\displaystyle \|\hat f \|_{L^{\infty}({\mathbb R}^n)}\leq \|f\|_{L^1({\mathbb R}^n)}.

(iv) (Riemann-Lebesgue) We have that

\displaystyle \lim _{|\xi|\rightarrow +\infty} \hat f(\xi)=0. Continue reading

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DMat0101, Notes 2: Convolution, Dense subspaces and interpolation of operators

1. Convolutions and approximations to the identity

We restrict our attention to the Euclidean case {({\mathbb R}^n,\mathcal L,dx)}. As we have seen the space {L^1({\mathbb R}^n)} is a vector space; linear combinations of functions in {L^1({\mathbb R}^n)} remain in the space. There is however a `product’ defined between elements of {L^1({\mathbb R}^n)} that turns {L^1} into a Banach algebra. For {f,g\in L^1({\mathbb R}^n)} we define the convolution of {f*g} to be the function

\displaystyle (f*g)(x)=\int_{{\mathbb R}^n} f(y)g(x-y)dy = \int_{{\mathbb R}^n} g(y)f(x-y)dy.

Furthermore, using Fubini’s theorem to change the order of integration we can easily see that

\displaystyle  \begin{array}{rcl}  \|f*g\|_{L^1({\mathbb R}^n)}\leq \|f\|_{L^1({\mathbb R}^n)}\|g\|_{L^1({\mathbb R}^n)}. \end{array}

Thus for {f,g\in L^1({\mathbb R}^n)} we have that their convolution {f*g} is again an element of {L^1({\mathbb R}^n)}. Note that the previous estimate is the main difficulty in showing that {(L^1({\mathbb R}^n),*)} is a Banach algebra.

More generally, the convolution of {f\in L^p({\mathbb R}^n)}, {1\leq p \leq +\infty}, and {g\in L^1({\mathbb R}^n)}, is a well defined element of {L^p({\mathbb R}^n)} and we have that

\displaystyle  \|f*g\|_{L^p({\mathbb R}^n)}\leq \|f\|_{L^p({\mathbb R}^n)}\|g\|_{L^1({\mathbb R}^n)}. \ \ \ \ \ (1)

Continue reading

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DMat0101, Notes 1: Quick review of measure theory

0. About these notes

The notes that will follow are meant to be a companion to the Harmonic Analysis course that I’m giving this semester at IST. These notes are inspired, influenced and sometimes shamelessly copied from books, lecture notes of other people, research papers and online material. The whole idea and structure of the course and, in particular, the use of the blog as a general platform of communication and interaction in relevance to the course is highly influenced by similar efforts of the other people and, especially, from Terence Tao’s blog as well as his lecture notes. Be sure to check the originals!

1. Introduction and notations

As mentioned in the overview of the course, we will be mainly concerned with operators acting on certain function spaces, or even spaces of more rough objects such as measures or distributions. Typically we will want to study the mapping properties of such an operator, that is whether it maps one function space to another and so on. A typical estimate in this context is of the form

\displaystyle \|Tf\|_Y\leq C \|f\|_X, Continue reading

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Course Announcement DMat0101: Harmonic Analysis, PhD course at IST

This semester, starting the week 14-21 of February, I’m giving a course in Harmonic Analysis on the PhD level as part of the Doctoral Program in Mathematics at the Department of Mathematics at IST. I will use this blog in order to coordinate the course, post comments and notes, as well as a means of communicating with whoever is interested in following the course. For this I expect the comment function of the blog to play a central role and give everyone an easy way to comment on the content of a lecture or ask questions related to the course in general.

Continue reading

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The maximal function along a polynomial curve; effective dimension bounds.

In this post I will try to give a description of an older result of mine that studies the {L^2\rightarrow L^2} operator norm of the maximal function along a polynomial curve. The relevant paper can be found here. The main object of study in this paper is the maximal operator

\displaystyle \mathcal{M}_P(f)(x):=\sup_{\epsilon>0}\frac{1}{2\epsilon} \int_{|t|\leq \epsilon} |f(x_1-t,x_2-t^2,\ldots,x_d-t^d)|dt.

It was known since the seventies that this operator is bounded on {L^p} for {1<p<\infty}. I was however interested in getting some effective bounds for the operator norm, at least on {L^2}. In fact it is possible to do that:

Theorem 1 (Parissis, 2010) There is a numerical constant {c>0} such that

\displaystyle 	\|\mathcal{M}_P(f)\|_{L^2({\mathbb R}^d)} \leq c \log d\ \| f\|_{L^2({\mathbb R}^d)}.

In this post we will content ourselves to proving a slightly weaker estimate with linear (instead of logarithmic) growth in {d}. This will serve presenting the main ideas and techniques involved in the proof while keeping things as simple as possible. I will however give some clues on how to move from the linear dependence to the logarithmic one without presenting too many details. Of course the reader can always consult the original paper where all the details are presented.

The methods and ideas in this paper are somehow a mix originating in two independent investigations. The first is concerned with the dimension dependence of the operator norm of the maximal function. It was first Stein that observed that the Hardy-Littlewood maximal function associated with the Euclidean ball function is bounded on {L^p} with norm bounds that do not depend on the dimension. The second area of research has to do with the boundedness properties of maximal functions (and singular integrals) along lower dimensional varieties. The operator under study is such an example. However since here I am interested in good dimensional constants for the the norm of such an operator, tools from the first area of research will be used. I will try to give a short overview of these two areas. I will then try to describe how Bourgain’s ideas for the study of the standard maximal function can be used together with some new ones in order to get a good operator bound for the maximal function along a polynomial curve. Continue reading

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Szemerédi’s theorem, frequent hypercyclicity and multiple recurrence

My co-author George Costakis and I have recently uploaded to arxiv our paper “Szemerédi’s theorem, frequent hypercyclicity and multiple recurrence”. As I’m invited to talk about this subject next month, I will try to give here a general overview of the paper, the notions therein and the main ideas involved in the proofs. Our main objective in this paper is to relate some notions in linear dynamics to more classical notions from topological dynamics. In particular we show that frequently Cesàro hypercyclic operators are necessarily topologically multiply recurrent. The main tool we use to prove this result is Szemerédi’s theorem on arithmetic progressions in sets of positive density. In order to motivate this theorem, I will have to define many standard notions from linear dynamics as well as corresponding notions from topological dynamics. Before discussing the main result and (some of) its applications, I will try to give a picture of hypercyclic operators and their properties, as well as examples of `natural’ operators which are hypercyclic.

— 1. Introduction: notions of hypercyclicity —

First of all, I will review some basic notions from linear dynamics that will be quite central throughout the exposition. I refer the reader to the excellent book of Bayart and Matheron (Bayart and Matheron, 2009) where most of this material is drawn from anyways. We will state several classical results here omitting the proof. If no other reference is given, this means the proof can be found in (Bayart and Matheron, 2009).

— 1.1. Hypercyclic operators —

We will work on a separable Banach space {X} over {{\mathbb R}} or {{\mathbb C}}. We will always use the symbol {T:X\rightarrow X} to denote a bounded linear operator acting on {X}. In what follows I will just write {X}, {T}, without any further comment, assuming always that these symbols have the meaning described above.

The most central notion in linear dynamics is that of hypercyclicity. Continue reading

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The Rudin (Hardy-Littlewood) Conjecture, Notes 3: Omissions.

1. The set of squares does not contain arbitrarily long arithmetic progressions.

As we have mentioned in the beginning of this discussion, Rudin was originally interested (among other things) in the number of terms the set of squares {S} has in arithmetic progressions. For this we recall the definition of the quantity {\alpha_N(E)} for a set {E\subset \mathbb Z}. For {a,b\in\mathbb Z} and {N\in\mathbb N} we define {\alpha_E(N,a,b)} to be the number of terms of the set {E} contained in the arithmetic progression

\displaystyle  \begin{array}{rcl}  	 a+b,a+2b,\ldots,a+Nb. \end{array}

We then define {\alpha_E(N){\stackrel{\mathrm {def}}{=}} \sup_{a,b\in\mathbb Z} \alpha_E(N,a,b)}. That is, {\alpha_E(N)} is the maximum number of terms of {E} contained in arithmetic progressions of length {N}.

For the set of squares {S=\{1,2^2,3^2\ldots\}} Erdös conjectured that we must have {\alpha_S(N)=o(N)}. This is was first proved by Szemerédi:

Theorem 1 (Szemerédi) We have that {\alpha_S(N)=o(N)}.

Continue reading

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