Iosevich A., Liflyand E. Decay of the Fourier Transform: Analytic and Geometric Aspects

Basel: Birkhäuser, 2014. — 226 p.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.​Basic Properties of the Fourier TransformOscillatory Integrals
The Fourier Transform of Convex and Oscillating Functions
The Fourier Transform of a Radial FunctionL 2 -average Decay of the Fourier Transform of a Characteristic Function of a Convex Set
L 1 -average Decay of the Fourier Transform of a Characteristic Function of a Convex Set
Geometry of the Gauss Map and Lattice Points in Convex Domains
Average Decay Estimates for Fourier Transforms of Measures Supported on Curves