Definition:Multiplication of Distribution by Smooth Function
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Definition
Let $d \in \N$ be a natural number.
Let $\alpha \in \map {C^\infty} {\R^d}$ be a smooth function.
Let $T \in \map {\DD'} {\R^d}$ be a distribution.
Then the multiplication of a distribution by a smooth function $\alpha T$ is defined by:
- $\alpha \map T \phi := \map T {\alpha \phi}$
where $\phi \in \map \DD {\R^d}$ is a test function.
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 6.4$: A glimpse of distribution theory. Multiplication by $C^\infty$ functions