Principal-value reciprocal distribution ($\mathrm{pv}(1/x)$)

The Cauchy principal valuedefines a distribution. Near zero the odd constant contribution cancels, and the remaining numerator is $O(x)$. It satisfies $x\mathrm{pv}(1/x)=1$. Hence all solutions of $xv=1$ are $v=\mathrm{pv}(1/x)+c{\delta }_0$ by the kernel of multiplication by a coordinate.

For a continuous even function $f$ on $\mathbb{R}$, the Cauchy principal valuedefines a distribution of order of a distribution at most one. For test functions supported in $[-R,R]$, the absolute value is at most $2R{\max }_{[0,R]}|f|\Vert {\varphi }^{\prime }{\Vert }_{\infty }$, since the numerator is bounded by $2x\Vert {\varphi }^{\prime }{\Vert }_{\infty }$. When $\varphi (0)=0$, the expression equals the absolutely convergent ordinary integral $\int f\varphi /x$.

It is a regular distribution exactly when ${\int }_0^r|f(x)|/xdx<\infty$ 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 ${\int }_1^{\infty }ds/s$.

Target: L1-normalized dilation of a distribution. Parent: Dilation of a distribution. Intended file: distribution-theory.bigb.