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Dilation of a distribution

{title2=$\langle u_\varepsilon,\phi\rangle=\langle u,\phi(\varepsilon\,\cdot)\rangle$}

= L1-normalized distributional dilation
{synonym}

For $\varepsilon>0$ and $u\in\mathcal D'(\mathbb R^n)$, define $u_\varepsilon$ by the displayed pairing. If $u$ is a <regular distribution>, this agrees with $u_\varepsilon(x)=\varepsilon^{-n}u(x/\varepsilon)$ and preserves its <L1 space> norm. With the convention $\delta_tu(x)=u(tx)$ for <dilation of a distribution>, it equals $\varepsilon^{-n}\delta_{1/\varepsilon}u$.

If $u\in L^1(\mathbb R^n)$, the <dominated convergence theorem> gives
$$
\langle u_\varepsilon,\phi\rangle=\int u(y)\phi(\varepsilon y)\,dy
\longrightarrow\phi(0)\int u,
\qquad u_\varepsilon\to\left(\int u\right)\delta_0.
$$
The constant <distribution> $u=1$ instead gives $u_\varepsilon=\varepsilon^{-n}$ and has no limit in <weak convergence of distributions>.

``
Target: Restriction of a distribution. Parent: Distribution (mathematical analysis). Intended file: distribution-theory.bigb.
``

L1-normalized dilation of a distribution

Past exam of the distribution theory course of the University of Oxford / 2023 / 2 / a / iii / Solution

codex/l1-normalized-distributional-dilation

L1-normalized dilation of a distribution ( $⟨ u_{ε}, ϕ ⟩ = ⟨ u, ϕ (ε \cdot)⟩$ )

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L1-normalized dilation of a distribution (⟨uε​,ϕ⟩=⟨u,ϕ(ε⋅)⟩)