Definition:Fourier Transform of Tempered Distribution

Definition
Let $T \in \map {\SS'} \R$ be a tempered distribution.

Let $\phi \in \map \SS \R$ be a Schwartz test function.

The Fourier transform $\hat T$ of (the tempered distribution) $T$:


 * $\hat T \in \map {\SS'} \R$

is defined as:


 * $\forall \phi \in \map \SS \R: \map {\hat T} \phi := \map T {\hat \phi}$

Also see

 * Fourier Transform of Tempered Distribution is Well-Defined
 * Fourier Transform of Tempered Distribution is Tempered Distribution