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

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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.
In particular: 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.
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$\blacksquare$


Sources

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