Decay of the Fourier Transform

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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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.​

• Fourier transform
• bounded variation
• curvature
• decay rate
• spherical average

“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)

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

  Alex Iosevich

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

  Elijah Liflyand

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