Differentiable Function as Distribution
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Theorem
Let $T \in \map {\DD'} \R$ be a distribution.
Let $f : \R \to \R$ be a continuously differentiable real function.
Suppose $T_f$ is a distribution identified with $f$.
Then $T_f' = T_{f'}$.
Proof
Let $\phi \in \map \DD \R$ be a test function with a support on $\closedint a b$.
Then:
\(\ds \map {T'_f} \phi\) | \(=\) | \(\ds -\map {T_f} {\phi'}\) | Definition of Distributional Derivative | |||||||||||
\(\ds \) | \(=\) | \(\ds -\int_{-\infty}^\infty \map f x \map {\phi'} x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\int_a^b \map f x \map {\phi'} x \rd x\) | Definition of Test Function, Integral of Compactly Supported Function | |||||||||||
\(\ds \) | \(=\) | \(\ds -\paren {\map f b \map \phi b - \map f a \map \phi a} + \int_a^b \map \phi x \map {f'} x \rd x\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_a^b \map \phi x \map {f'} x \rd x\) | Definition of Test Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {T_{f'} } \phi\) |
$\blacksquare$
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 6.2$: A glimpse of distribution theory. Derivatives in the distributional sense