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Birkhäuser

Decay of the Fourier Transform

Analytic and Geometric Aspects

  • Book
  • © 2014

Overview

Authors:
  1. Alex Iosevich

    1. Department of Mathematics, University of Rochester, Rochester, USA

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    You can also search for this author in PubMed   Google Scholar

  2. Elijah Liflyand

    1. Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • Only book where the decay rate of the Fourier transform is the dominant theme
  • Systematic examination of the concepts
  • Focus on interaction between the analytic and geometric approaches of Fourier theory?

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-xii
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  2. Preliminaries

    1. Front Matter

      Pages 1-1
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    2. Introduction

      • Alex Iosevich, Elijah Liflyand
      Pages 3-10

    3. Basic Properties of the Fourier Transform

      • Alex Iosevich, Elijah Liflyand
      Pages 11-16

  3. Analytic (and Geometric) Aspects

    1. Front Matter

      Pages 17-17
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    2. Oscillatory Integrals

      • Alex Iosevich, Elijah Liflyand
      Pages 19-46

    3. The Fourier Transform of Convex and Oscillating Functions

      • Alex Iosevich, Elijah Liflyand
      Pages 47-91

    4. The Fourier Transform of a Radial Function

      • Alex Iosevich, Elijah Liflyand
      Pages 93-126

  4. Geometric (and Analytic) Aspects

    1. Front Matter

      Pages 127-127
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    2. L 2-average Decay of the Fourier Transform of a Characteristic Function of a Convex Set

      • Alex Iosevich, Elijah Liflyand
      Pages 129-135

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

      • Alex Iosevich, Elijah Liflyand
      Pages 137-160

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

      • Alex Iosevich, Elijah Liflyand
      Pages 161-171

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

      • Alex Iosevich, Elijah Liflyand
      Pages 173-203

  5. Back Matter

    Pages 205-222
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Keywords

About this book

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.​

Reviews

“The overall presentation is clear and the reader is carefully guided through this territory which is marked by a beautiful blend of ideas and techniques from analysis, geometry and also number theory.” (Wien R. Steinbauer, Monatshefte für Mathematik, Vol. 185, 2018)


“It is an introduction to current topics in Fourier analysis, to be read and appreciated by mathematicians … . the book is carefully designed and well written. It does a very good job of familiarizing the reader with the relevant techniques and results, relying on a beautiful interplay of analysis, geometry and number theory.” (Hartmut Führ, Mathematical Reviews, January, 2016)

Authors and Affiliations

  • Department of Mathematics, University of Rochester, Rochester, USA

    Alex Iosevich

  • Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel

    Elijah Liflyand

Bibliographic Information

  • Book Title: Decay of the Fourier Transform

  • Book Subtitle: Analytic and Geometric Aspects

  • Authors: Alex Iosevich, Elijah Liflyand

  • DOI: https://doi.org/10.1007/978-3-0348-0625-1

  • Publisher: Birkhäuser Basel

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer Basel 2014

  • Hardcover ISBN: 978-3-0348-0624-4Published: 10 October 2014

  • eBook ISBN: 978-3-0348-0625-1Published: 01 October 2014

  • Edition Number: 1

  • Number of Pages: XII, 222

  • Number of Illustrations: 3 b/w illustrations, 2 illustrations in colour

  • Topics: Analysis, Fourier Analysis

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