Return to Revisions

The following basic result needs to be quoted on these matters of analyticity: the Paley-Wiener-Schwartz theorem gives a characterization of distributions with compact support. Let $u$ be a tempered distribution ; $u$ is compactly supported in a ball of center $0$ and radius $R$ if an only if $\hat u$ is an entire function such that $$\exists C_0, \exists N_0,\forall \zeta\in \mathbb C^n,\quad \vert\hat u(\zeta)\vert\le C_0(1+\vert \zeta\vert)^{N_0}e^{R\vert\Im\zeta\vert}. $$ Something analogous allows a characterization of $C^\infty$ functions with compact support.