Definition:Fourier Transform of Tempered Distribution

Definition

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

Let $\map \SS \R$ be the Schwartz space.

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

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 6.5$: A glimpse of distribution theory. Fourier transform of (tempered) distributions