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, $\map x 0 = x_0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \frac {\map \phi x \map {\delta_n} {x - x_0} } {\size {\map {f'} {x_0 } } } \rd x\) |
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$\blacksquare$
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