= Even-weight principal-value distribution
{title2=$\operatorname{pv}(f(x)/x)$}
For a continuous even <function> $f$ on $\mathbb R$, the <Cauchy principal value>
$$
\left\langle\operatorname{pv}\frac{f(x)}x,\phi\right\rangle
=\int_0^\infty f(x)\frac{\phi(x)-\phi(-x)}x\,dx
$$
defines a <distribution> of <order of a distribution> at most one. For <test functions> supported in $[-R,R]$, the absolute value is at most $2R\max_{[0,R]}|f|\,\|\phi'\|_\infty$, since the numerator is bounded by $2x\|\phi'\|_\infty$. When $\phi(0)=0$, the expression equals the absolutely convergent ordinary integral $\int f\phi/x$.
It is a <regular distribution> exactly when $\int_0^r|f(x)|/x\,dx<\infty$ for some $r>0$. Necessity follows by testing away from zero: any representing <locally integrable function> must equal $f/x$ <almost everywhere> off zero. Sufficiency follows from absolute convergence. Continuity with $f(0)=0$ alone does not suffice: $f(x)=1/\log(1/|x|)$ near zero gives the divergent integral $\int_1^\infty ds/s$.
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Target: L1-normalized dilation of a distribution. Parent: Dilation of a distribution. Intended file: distribution-theory.bigb.
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