Definition:Derivative of Tempered Distribution
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Definition
Let $\phi \in \map \SS \R$ be a Schwartz test function.
Let $T \in \map {\SS'} \R$ be a tempered distribution.
The derivative of tempered distribution $\ds \dfrac {\d T} {\d x} \in \map {\SS'} \R$ is defined by:
- $\map {\dfrac {\d T} {\d x}} \phi := - \map T {\dfrac {\d \phi} {\d x}}$
Also see
- Results about distributional derivatives can be found here.
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