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