Composition of Dirac Delta Distribution with Function with Simple Zero/Proof 1

Theorem

Let $\delta \in \map {\DD'} \R$ be the Dirac delta distribution.

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

Let $f : \R \to \R$ be a real function with a simple zero at $x_0$.

Let $f$ be strictly monotone.

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

Then in the distributional sense it holds that:

$\ds \map \delta {\map f x} = \frac {\map \delta {x - x_0}}{\size {\map {f'} {x_0}} }$

which can be interpreted as:

$\ds \int_{-\infty}^\infty \map \delta {\map f x} \map \phi x \rd x = \frac {\map \phi {x_0}}{\size {\map {f'} {x_0}} }$

which more strictly means that:

$\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map {\delta_n} {\map f x} \map \phi x \rd x = \frac {\map \phi {x_0}}{\size {\map {f'} {x_0}} }$

Proof

Suppose $\map {f'} {x_0} > 0$.

Then:

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

\(=\)

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

Derivative of Inverse Function, Integration by Substitution, $y = \map f x$

\(\ds \)

\(=\)

\(\ds \frac {\map \phi {x_0} }{\map {f'} {x_0} }\)

Definition of Delta Sequence, $\map x 0 = x_0$

\(\ds \)

\(=\)

\(\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \frac {\map \phi x \map {\delta_n} {x - x_0} }{\map {f'} {x_0} } \rd x\)

Suppose $\map {f'} {x_0} < 0$.

Then:

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

\(=\)

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

Derivative of Inverse Function, Integration by Substitution, $y = \map f x$

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

\(\ds \frac {\map \phi {x_0} }{- \map {f'} {x_0} }\)

Definition of Delta Sequence, $\map x 0 = x_0$

\(\ds \)

\(=\)

\(\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \frac {\map \phi x \map {\delta_n} {x - x_0} }{- \map {f'} {x_0} } \rd x\)

Altogether:

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

\(=\)

\(\ds \frac {\map \phi {x_0} }{ \size { \map {f'} {x_0} } }\)

Definition of Delta Sequence, $\map x 0 = x_0$

\(\ds \)

\(=\)

\(\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \frac {\map \phi x \map {\delta_n} {x - x_0} }{ \size {\map {f'} {x_0 } } } \rd x\)

Further research is required in order to fill out the details. Theorem conditions here are stronger than usually found in literature to keep us on the safe side. The situtation may change as we learn more. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by finding out more. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Research}} from the code.

$\blacksquare$

Sources

1998: Ram Prakash Kanwal: Generalized Functions: Theory and Technique (2nd ed.): Chapter $3$. Additional Properties of Distributions $3.1$ Transformation Properties of the Delta Distribution