Definition:Fourier Transform of Tempered Distribution
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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