Asymptotics of Fourier Transforms

Abstract

In Chapter 3 we demonstrated that the Fourier transform $\tilde f\left( \omega \right)$ of a smooth function f(t) rapidly decays to zero as ω → ∞. However, smoothness is rare in natural phenomena and one often encounters processes that are either discontinuous or violate the smoothness assumption in other ways. Such phenomena include, for example, shock fronts generated by large amplitude acoustic waves, ocean waves, or desert dunes with their characteristic sharp crests. These and many other examples explain the importance of the Fourier analysis of nonsmooth processes.