Browsing by Subject "Fourier analysis"

In this thesis we study the article by J. Bourgain and C. Demeter called A study guide for the l^2 decoupling theorem. In particular, we hope to give an in detail exposition to certain results from the aforementioned research article so that this text combined with the master’s thesis On the l^2 decoupling theorem by Jaakko Sinko covers the l^2 decoupling theorem comprehensively in the range 2 ≤ p ≤ 2n/(n−1). The results in this text also self-sufficiently motivate the use of the extension operator and explain why it is possible to prove linear decouplings with multilinear estimates. We begin the thesis by giving the basic notation and highlighting some useful results from analysis and linear algebra that are later used in the thesis. In the second chapter we introduce and prove a certain multilinear Kakeya inequality, which asserts an upper bound for the overlap of neighbourhoods of nearly axis parallel lines in R^n that point in different directions. In the next chapter this is applied to prove a multilinear cube inflation inequality, which is one of the main mechanisms in the proof of the l^2 decoupling theorem. In the fourth chapter we study two forms of linear decoupling. One that is defined by an extension operator and one that defined via Fourier restriction. The main result of this chapter is that the former is strong enough to produce decoupling inequalities that are of the latter form. The fifth chapter is reserved for comparing linear and multilinear decouplings. Here we use the main result of the previous chapter to prove that multilinear estimates can produce linear decouplings, if the lower dimensional decoupling constant is somehow contained. This paves the way for the induction proof of the l^2 decoupling theorem.

This thesis surveys the vast landscape of uncertainty principles of the Fourier transform. The research of these uncertainty principles began in the mid 1920’s following a seminal lecture by Wiener, where he first gave the remark that condenses the idea of uncertainty principles: "A function and its Fourier transform cannot be simultaneously arbitrarily small". In this thesis we examine some of the most remarkable classical results where different interpretations of smallness is applied. Also more modern results and links to active fields of research are presented.We make great effort to give an extensive list of references to build a good broad understanding of the subject matter.Chapter 2 gives the reader a sufficient basic theory to understand the contents of this thesis. First we talk about Hilbert spaces and the Fourier transform. Since they are very central concepts in this thesis, we try to make sure that the reader can get a proper understanding of these subjects from our description of them. Next, we study Sobolev spaces and especially the regularity properties of Sobolev functions. After briefly looking at tempered distributions we conclude the chapter by presenting the most famous of all uncertainty principles, Heisenberg’s uncertainty principle.In chapter 3 we examine how the rate of decay of a function affects the rate of decay of its Fourier transform. This is the most historically significant form of the uncertainty principle and therefore many classical results are presented, most importantly the ones by Hardy and Beurling. In 2012 Hedenmalm gave a beautiful new proof to the result of Beurling. We present the proof after which we briefly talk about the Gaussian function and how it acts as the extremal case of many of the mentioned results.In chapter 4 we study how the support of a function affects the support and regularity of its Fourier transform. The magnificent result by Benedicks and the results following it work as the focal point of this chapter but we also briefly talk about the Gap problem, a classical problem with recent developments.Chapter 5 links density based uncertainty principle to Fourier quasicrystals, a very active field of re-search. We follow the unpublished work of Kulikov-Nazarov-Sodin where first an uncertainty principle is given, after which a formula for generating Fourier quasicrystals, where a density condition from the uncertainty principle is used, is proved. We end by comparing this formula to other recent formulas generating quasicrystals.