Distributional Derivatives of Dirac Delta Distribution do not Vanish

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.

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 $

$=$

$\ds \paren {-1}^n \map \delta {\sum_{k \mathop = 0}^n \binom n k \paren {\dfrac {\d^k} {\d x^k} x^n} \phi^{\paren {n - k} } }$

$\ds $

$=$

$\ds \paren {-1}^n \binom n n n! \map \phi 0$

$\ds $

$\ne$

$\ds 0$

$\blacksquare$

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