Decay of the Fourier Transform: Analytic and Geometric Aspects

ISBN-13: 9783034806244 / Angielski / Twarda / 2014 / 222 str.

Iosevich

Opis

The Plancherel formula says that the L DEGREES2 norm of the function is equal to the L DEGREES2 norm of its Fourier transform. This implies that at least on average, the Fourier transform of an L DEGREES2 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 DEGREES2 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.​

Recenzje

"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)

Spis treści

Foreword.- Introduction.- Chapter 1. Basic properties of the Fourier transform.- Chapter 2. Oscillatory integrals and Fourier transforms in one variable.- Chapter 3. The Fourier transform of an oscillating function.- Chapter 4. The Fourier transform of a radial function.- Chapter 5. Multivariate extensions.- Appendix.- Bibliography.​

Blurb

The Plancherel formula says that the L² norm of the function is equal to the L² norm of its Fourier transform. This implies that at least on average, the Fourier transform of an L² 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 L² 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.