Principal Value of One over x is Distribution

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Theorem

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

Let $T : \map \DD \R \to \C$ be a mapping such that:

$\forall \phi \in \map \DD \R : \map T \phi = \PV \frac {\map \phi x} x \rd x : = \lim_{\epsilon \mathop \to 0} \int_{\size x \mathop > \epsilon} \frac {\map \phi x} x \rd x$

where $PV$ denotes the Cauchy principal value.


Then $T$ is a distribution.


Proof

Let $\phi \in \map \DD \R$ be a test function with a support on $\closedint {-a} a$.

Then:

\(\ds \map \phi x\) \(=\) \(\ds \map \phi 0 + \int_0^x \dfrac {\d \map \phi x} {\d x} \rd x\)
\(\ds \) \(=\) \(\ds \map \phi 0 + x \int_0^1 \dfrac {\d \map \phi {t x} } {\d \paren{t x} } \rd t\)

Furthermore:

\(\ds \int_{a \mathop \ge \size x \mathop > \epsilon} \frac 1 x \rd x\) \(=\) \(\ds \int_\epsilon^a \frac 1 x \rd x + \int_{-a}^{-\epsilon} \frac 1 x \rd x\)
\(\ds \) \(=\) \(\ds \int_\epsilon^a \frac 1 x \rd x + \int_a^\epsilon \frac 1 x \rd x\)
\(\ds \) \(=\) \(\ds 0\)


Existence of the limit

\(\ds \int_{\size x \mathop > \epsilon} \frac {\map \phi x} x \rd x\) \(=\) \(\ds \int_{a \mathop \ge \size x \mathop > \epsilon} \frac {\map \phi x} x \rd x\)
\(\ds \) \(=\) \(\ds \map \phi 0 \int_{a \mathop \ge \size x \mathop > \epsilon} \frac 1 x \rd x + \int_{a \mathop \ge \size x \mathop > \epsilon} \rd x \int_0^1 \dfrac {\d \map \phi {t x} }{\d \paren {t x} } \rd t\)
\(\ds \) \(=\) \(\ds 0 + \int_{a \mathop \ge \size x \mathop > \epsilon} \rd x \int_0^1 \dfrac {\d \map \phi {t x} }{\d \paren {t x} } \rd t\)
\(\ds \) \(=\) \(\ds \int_{a \mathop \ge \size x \mathop > \epsilon} \rd x \int_0^1 \dfrac {\d \map \phi {t x} }{\d \paren {t x} } \rd t\)

Since $\phi \in \map \DD \R$, the integral exists for any $\epsilon$.

Hence, the limit exists.

Thus, we can rewrite $T$ as:

$\ds \map T \phi = \int_{-a}^a \int_0^1 \dfrac {\d \map \phi {t x} } {\d \paren {t x} } \rd t \rd x$


Linearity

Follows from Riemann Integral Operator is Linear Mapping.


Continuity

Let $\mathbf 0 : \R \to 0$ be a zero mapping.

Let $\sequence {\phi_n}_{n \mathop \in \N}$ be a sequence with support on $\closedint {-a} a$ such that it converges to $\mathbf 0$:

$\phi_n \stackrel \DD {\longrightarrow} \mathbf 0$

Then:

\(\ds \size {\map T {\phi_n} }\) \(=\) \(\ds \size {\int_{-a}^a \int_0^1 \dfrac {\d \map {\phi_n} {t x} } {\d \paren {t x} } \rd t \rd x}\)
\(\ds \) \(\le\) \(\ds 2a \cdot 1 \cdot \sup_{x \mathop \in \R} \size {\map {\phi_n'} x}\)

Take the limit $n \to \infty$.

Then:

$\map T {\mathbf 0} = 0$


$\blacksquare$


Sources