Composition of Dirac Delta Distribution with Function with Simple Zero/Corollary

Corollary to Composition of Dirac Delta Distribution with Function with Simple Zero
Dirac delta distribution is even:


 * $\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map {\delta_n} {-x} \map \phi x \rd x = \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map {\delta_n} x \map \phi x \rd x$

which can be abbreviated to:


 * $\map \delta {-x} = \map \delta x$

Proof
Let $\map \phi x \in \map \DD \R$ be a test function.

Let $\sequence {\map {\delta_n} x}_{n \mathop \in \N}$ be a delta sequence.

By definition:


 * $\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map {\delta_n} x \map \phi x \rd x = \map \phi 0$

Consider a delta sequence $\sequence {\map {\delta_n} {-x} }_{n \mathop \in \N}$.

This is a composition of $\sequence {\map {\delta_n} x}_{n \mathop \in \N}$ with the function $\map f x = - x$ with a simple zero at $x_0 = 0$.

By Composition of Dirac Delta Distribution with Function with Simple Zero:

Abusing the notation, this could be understood as:


 * $\ds \int_{-\infty}^\infty \map \delta {- x} \map \phi x \rd x = \int_{-\infty}^\infty \map \delta x \map \phi x \rd x$

or


 * $\map \delta {- x} = \map \delta x$