Decay of the Fourier Transform Analytic and Geometric Aspects [Hardcover]

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.Analytic and geometric properties of the underlying functions interact in a seamless symbiosis in this book, which examines the Decay of the Fourier Transform under various assumptions and circumstances. Goes far beyond the original L^2 setting.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.

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 f?r 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 F?hr, Mathematical Reviews, January, 2016)

The Plancherel formula says that the L