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:

\(\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map {\delta_n} {- x} \map \phi x \rd x\)

\(=\)

\(\ds \frac {\map \phi 0} {\size {-1} }\)

\(\ds \)

\(=\)

\(\ds \map \phi 0\)

\(\ds \)

\(=\)

\(\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map {\delta_n} x \map \phi x \rd x\)

Definition of Delta Sequence

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

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

$\blacksquare$

Sources

• 1998: Ram Prakash Kanwal: Generalized Functions: Theory and Technique (2nd ed.): Chapter $2$. The Schwartz-Sobolev Theory of Distributions $2.5$ Algebraic Operations on Distributions