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2014 | Buch

Kapitel lesen Erstes Kapitel lesen

Decay of the Fourier Transform

Analytic and Geometric Aspects

verfasst von: Alex Iosevich, Elijah Liflyand

Verlag: Springer Basel

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The Plancherel formula says that the L^2 norm of the function is equal to the L^2 norm of its Fourier transform. This implies that at least on average, the Fourier transform of an L^2 function decays at infinity. This book is dedicated to the study of the rate of this decay under various assumptions and circumstances, far beyond the original L^2 setting. Analytic and geometric properties of the underlying functions interact in a seamless symbiosis which underlines the wide range influences and applications of the concepts under consideration.

Inhaltsverzeichnis

Frontmatter

Preliminaries

Frontmatter

Chapter 0. Introduction

Abstract

This is a rather informal introduction, designed to describe the ideas and points of view of this book, rather than precise definition and calculations. All issues described below are treated in a precise matter later in the book. Our purpose is to show how various concepts arise, why they are important, and how they fit together. In other words, one should view this introduction as an informal microcosm of the book which can be read separately or as a launch pad into the rest of the treatise.

Alex Iosevich, Elijah Liflyand

Chapter 1. Basic Properties of the Fourier Transform

Abstract

In this chapter we define the Fourier transform and describe its basic properties. Since this part of the book is quite standard, we go through the material quickly with an eye on developments in the subsequent chapters.

Alex Iosevich, Elijah Liflyand

Analytic (and Geometric) Aspects

Frontmatter

Chapter 2. Oscillatory Integrals

Abstract

The method of stationary phase is the term typically applied to study of the integrals of the form $ \int_{\mathbb{R}^{d}}{e^{iRG(x)}{\psi(x)dx}}$ by studying properties of derivatives of the real or complex-valued phase function G(x) on the support of the cut-off $ \psi (x)$

Alex Iosevich, Elijah Liflyand

Chapter 3. The Fourier Transform of Convex and Oscillating Functions

Abstract

What may be referred to as an initial point for the subject of this chapter is Trigub’s result of the 1970’s on the asymptotics of the Fourier transform of a convex function (see, e.g., [198]).

Alex Iosevich, Elijah Liflyand

Chapter 4. The Fourier Transform of a Radial Function

Abstract

Spherical symmetry is a very interesting and important property of a function. Theorem 1.5 gives that if f(x) is radial (depending only on ‌x‌), then $ \hat{f} $ is radial too.

Alex Iosevich, Elijah Liflyand

Geometric (and Analytic) Aspects

Frontmatter

Chapter 5. L 2-average Decay of the Fourier Transform of a Characteristic Function of a Convex Set

Abstract

Let B be a bounded open set in ℝ^d. As we note in the introduction, it is a consequence of the classical method of stationary phase that if $ \partial{B} $ is sufficiently smooth and has everywhere non-vanishing Gaussian curvature, then

$$ |\hat{X}B(Rw)| \lesssim R ^{-{\frac{d+1}{2}}}$$

with constants independent of ω

Alex Iosevich, Elijah Liflyand

Chapter 6. L 1-average Decay of the Fourier Transform of a Characteristic Function of a Convex Set

Abstract

In the previous chapter we obtained optimal L ²-average decay under the assumption that the set B is bounded and has a convex boundary. We now turn our attention to obtaining more detailed understanding of this problem in the twodimensional setting.

Alex Iosevich, Elijah Liflyand

Chapter 7. Geometry of the Gauss Map and Lattice Points in Convex Domains

Abstract

In the previous two chapters, we have gained a significant amount of understanding about the L ^p-average decay for the Fourier transform of characteristic functions of convex sets and considered some applications to problems in lattice point counting and discrepancy theory. In this chapter we consider more elaborate applications of average decay in number theory where the discrepancy function needs to be estimated for almost every rotation instead of averaging over rotations in some L ^p-norm. This naturally leads us to the examination of certain maximal functions and as a result brings in some classical harmonic analysis that arises so often in the first part of this book.

Alex Iosevich, Elijah Liflyand

Chapter 8. Average Decay Estimates for Fourier Transforms of Measures Supported on Curves

Abstract

The previous three chapters dealt with average decay for subsets of ℝ^d, which quickly reduces to the problem of average decay of Fourier transforms of measures supported on surfaces of co-dimension one. In this chapter we address the issue of Fourier transform of average decay of measures supported on curves in ℝ^d with some tantalizing connection with the classical restriction theory.

Alex Iosevich, Elijah Liflyand

Backmatter

Titel: Decay of the Fourier Transform
verfasst von: Alex Iosevich
Elijah Liflyand
Verlag: Springer Basel
Electronic ISBN: 978-3-0348-0625-1
Print ISBN: 978-3-0348-0624-4
DOI: https://doi.org/10.1007/978-3-0348-0625-1