Composition of Dirac Delta Distribution with Function with Simple Zero/Proof 2
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
Let $H$ be the Heaviside step function.
Let $T \in \map \DD \R$ be a distribution associated with $\map H {\map f x}$:
- $\ds T = T_{\map H {\map f x}}$
We have that:
\(\ds \map H {\map f x}\) | \(=\) | \(\ds \begin{cases} \map H {x - x_0} & : \forall x \in \R : \map {f'} x > 0 \\ 1 - \map H {x - x_0} & : \forall x \in \R : \map {f'} x < 0 \end{cases}\) |
Taking the derivative of the left hand side yields:
\(\ds \dfrac \d {\d x} \map H {\map f x}\) | \(=\) | \(\ds \dfrac {\d \map f x} {\d x} \dfrac {\d \map H {\map f x} } {\d \map f x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {f'} x \dfrac {\d \map H {\map f x} } {\d \map f x}\) |
Taking the derivative of the right hand side yields:
- $\forall x \ne x_0 : \dfrac \d {\d x} \map H {\map f x} = 0$
Furthermore:
- $\forall x \in \R : \map {f'} x > 0 : \paren {\map f {x_0^+} = 1} \land \paren {\map f {x_0^-} = 0}$
- $\forall x \in \R : \map {f'} x < 0 : \paren {\map f {x_0^+} = 0} \land \paren {\map f {x_0^-} = 1}$
By Jump Rule the right hand side reads:
\(\ds T_{ \map H {\map f x} }'\) | \(=\) | \(\ds \begin{cases} \delta_{x_0} & : \forall x \in \R : \map {f'} x > 0 \\ -\delta_{x_0} & : \forall x \in \R : \map {f'} x < 0 \end{cases}\) |
Define the composite Dirac delta distribution according to Distributional Derivative of Heaviside Step Function.
We have that:
- $T_{\map H x}' = \delta_0 = \delta_x$
Then:
\(\ds T_{\map H {\map f x} }'\) | \(=\) | \(\ds T_{\map {f'} x \frac {\d \map H {\map f x} } {\d \map f x} }\) | Derivative of Composite Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {f'} x \delta_{\map f x}\) | Multiplication of Distribution induced by Locally Integrable Function by Smooth Function |
Hence:
\(\ds \forall x \in \R : \map {f'} x\) | \(>\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \size {\map {f'} x} \delta_{\map f x}\) | \(=\) | \(\ds \delta_{x_0}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \delta_{\map f x}\) | \(=\) | \(\ds \frac {\delta_{x_0} } {\size {\map {f'} x} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\delta_{x_0} } {\size {\map {f'} {x_0} } }\) | Product of Smooth Function and Dirac Delta Distribution |
and:
\(\ds \forall x \in \R : \map {f'} x\) | \(<\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds -\size {\map {f'} x} \delta_{\map f x}\) | \(=\) | \(\ds -\delta_{x_0}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \delta_{\map f x}\) | \(=\) | \(\ds \frac {\delta_{x_0} } {\size {\map {f'} x} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\delta_{x_0} } {\size {\map {f'} {x_0} } }\) | Product of Smooth Function and Dirac Delta Distribution |
$\blacksquare$
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