Sobolev Spaces

Abstract

The degree of differentiability of a function u with compact support cannot easily be seen from the Fourier transform $\mathcal{F}\ u.$ For instance, there is a difference of more than n between the exponents in the estimates for $\mathcal{F}\ u.$ in Theorems 18.1 and 18.6, which are necessary and sufficient, respectively, for u to be of class C ^k. The question how to decide by inspection whether a function is the Fourier transform of a bounded continuous function is very difficult to answer in general.