Distributional Derivatives of Dirac Delta Distribution do not Vanish
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Theorem
Let $\delta \in \map {\DD'} \R$ be the Dirac delta distribution.
Then for any $n \in \N$ the distributional derivative $\delta^{\paren n}$ does not vanish.
Proof
Let $\phi \in \map \DD \R$ be a test function such that $\map \phi 0 \ne 0$.
Then:
- $\forall n \in \N : \forall x \in \R : x^n \phi \in \map \DD \R$
By the definition of the distributional derivative:
\(\ds \map {\delta^{\paren n} } {x^n \phi}\) | \(=\) | \(\ds \paren {-1}^n \map \delta {\paren {x^n \phi}^{\paren n} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^n \map \delta {\sum_{k \mathop = 0}^n \binom n k \paren {\dfrac {\d^k} {\d x^k} x^n} \phi^{\paren {n - k} } }\) | Leibniz's Rule in One Variable | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^n \binom n n n! \map \phi 0\) | Nth Derivative of Nth Power, Definition of Dirac Delta Distribution | |||||||||||
\(\ds \) | \(\ne\) | \(\ds 0\) |
$\blacksquare$
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