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...