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\begin{document}

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\section*{extension of distributions}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{SolutionSpaceOfPointExtensions}{Point-extensions}\dotfill \pageref*{SolutionSpaceOfPointExtensions} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{powers_of_feynman_propagators}{Powers of Feynman propagators}\dotfill \pageref*{powers_of_feynman_propagators} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Given a suitable [[subspace]] inclusion $X \hookrightarrow \hat X$ and a [[distribution]] $u$ on $X$, then an \emph{extension} of $u$ to $\hat X$ is a distribution $\hat u$ on $\hat X$ whose [[restriction of distributions]] to $X$ coincides with $u$. If $u$ comes from an ordinary [[smooth function]] or else if $u$ is properly understood as a [[generalized function]], then this corresponds to an ordinary [[extension]]

\begin{displaymath}
\itexarray{
    X &\overset{u}{\longrightarrow}& \mathbb{R}
    \\
    \downarrow & \nearrow_{\mathrlap{\hat u}}
    \\
    \hat X
  }
\end{displaymath}
Regarding that we the distributions are not the maps from the underlying space we need to replace pre-composition $X\to\tilde{X}$ by the appropriate [[pullback of distributions]] always defined say if $X\to\tilde{X}$ is a submersion; in the case of an [[open embedding]] (see \href{pullback+of+a+distribution#RestrictionOfDistributions}{this example}) this operation is the \emph{restriction of distributions}, the operation dual to the operation of extension by zero of test functions.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

For an inclusion of two open sets on a manifold $U\subset V$ there is an operator of \emph{extension by zero} $E_{V U}: C^\infty_0(U)\to C^\infty_0(V)$ where $E_{V U}(f)(x) = f(x)$ if $x\in U$ and $E_{U V}(f)(x) = 0$ otherwise. The \textbf{[[restriction of distributions]]} $\rho_{U V} : \mathcal{D}'(V)\to \mathcal{D}'(U)$ is then defined by

\begin{displaymath}
\langle \rho_{U V}(\phi), f\rangle = \langle \phi, E_{V U} f\rangle,
\,\,\,\,\,\,\,f\in C^\infty_0(U),\,\phi\in \mathcal{D}'(V).
\end{displaymath}
(\href{pullback+of+a+distribution#RestrictionOfDistributions}{this example}).

Now the diagram in the idea section makes sense in the following way: for an open embedding $X\hookrightarrow\hat{X}$, $\hat{u}\in\mathcal{D}'(\hat{X})$ \textbf{extends} $u\in \mathcal{D}'(X)$ if $\rho_{X\hat{X}}(\hat{u}) = u$.

\begin{defn}
\label{ExtensionOfDistributions}\hypertarget{ExtensionOfDistributions}{}
\textbf{([[extension of distributions]])}

Let $X \overset{\iota}{\subset} \hat X$ be an inclusion of [[open subsets]] of some [[Cartesian space]]. This induces the operation of [[restriction of distributions]]

\begin{displaymath}
\mathcal{D}'(\hat X) 
    \overset{\iota^\ast}{\longrightarrow}
  \mathcal{D}'(X)
  \,.
\end{displaymath}
Given a [[distribution]] $u \in \mathcal{D}'(X)$, then an \emph{[[extension]]} of $u$ to $\hat X$ is a distribution $\hat u \in \mathcal{D}'(\hat X)$ such that

\begin{displaymath}
\iota^\ast \hat u 
  \;=\;
  u
  \,.
\end{displaymath}
\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{SolutionSpaceOfPointExtensions}{}\subsubsection*{{Point-extensions}}\label{SolutionSpaceOfPointExtensions}

Of particular interest is the special case of [[extension of distributions]] to a single point $p \in \hat X$, hence where $X = \hat X \setminus \{p\}$ is the [[complement]] of that point.

Contrary to ordinary [[smooth functions]], distributions ([[generalized functions]]) in general have more than one extension to a point, where the freedom is in choosing a [[distribution with point support]] at that point, hence some [[derivative of a distribution|derivative]] of the [[delta distribution]] [[support of a distribution|supported]] at that point.

However, if one requires that the [[scaling degree of a distribution|scaling degree]] of the extended distribution at the given point is compatible with that of the original distribution, then there is only a [[finite set]] of possible extensions, parameterized by the [[coefficients]] of a finite number of derivatives of the delta-distribution (prop. \ref{SpaceOfPointExtensions} below).

In the construction of [[perturbative quantum field theories]] via [[causal perturbation theory]], where the (``[[operator-valued distributions|operator-valued]]'') distributions in questions are [[time-ordered product]] coefficients in the [[scattering matrix]] and where the point being extended to corresponds to the point where an [[interaction]] happens, this finite set of choices is identified with the [[renormalization|(``re''-)normalization]] freedom.

We discuss specifically the space of solutions of extending a distribution on the [[complement]] $\mathbb{R}^n \setminus \{0\}$ of the origin inside a [[Cartesian space]] to the full space $\mathbb{R}^n$.

\begin{prop}
\label{ExtensionUniqueNonPositiveDegreeOfDivergence}\hypertarget{ExtensionUniqueNonPositiveDegreeOfDivergence}{}
\textbf{(unique [[extension of distributions]] with negative [[degree of divergence of a distribution|degree of divergence]])}

For $n \in \mathbb{N}$, let $u \in \mathcal{D}'(\mathbb{R}^n \setminus \{0\})$ be a [[distribution]] on the [[complement]] of the origin, with [[negative number|negative]] [[degree of divergence of a distribution|degree of divergence]] at the origin

\begin{displaymath}
deg(u) \lt 0
  \,.
\end{displaymath}
Then $u$ has a \emph{unique} [[extension of distributions]] $\hat u \in \mathcal{D}'(\mathbb{R}^n)$ to the origin with the same degree of divergence

\begin{displaymath}
deg(\hat u) = deg(u)
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{BrunettiFredenhagen00}{Brunetti-Fredenhagen 00, theorem 5.2}, \hyperlink{Duetsch18}{Dütsch 18, theorem 3.35 a)})

\begin{proof}
Regarding uniqueness:

Suppose $\hat u$ and ${\hat u}^\prime$ are two extensions of $u$ with $deg(\hat u) = deg({\hat u}^\prime)$. Both being extensions of a distribution defined on $\mathbb{R}^n \setminus \{0\}$, this difference has [[support of a distribution|support]] at the origin $\{0\} \subset \mathbb{R}^n$. By \href{point-supported+distribution#PointSupportedDistributionsAreSumsOfDerivativesOfDeltaDistibutions}{this prop.} this implies that it is a linear combination of [[derivative of a distribution|derivatives]] of the [[delta distribution]] [[support of a distribution|supported]] at the origin:

\begin{displaymath}
{\hat u}^\prime - \hat u
  \;=\;
  \underset{
    {\alpha \in \mathbb{N}^n}
  }{\sum}
  c^\alpha \partial_\alpha \delta_0
\end{displaymath}
for constants $c^\alpha \in \mathbb{C}$. But by \href{scaling+degree+of+a+distribution#DerivativesOfDeltaDistributionScalingDegree}{this example} the [[degree of divergence of a distribution|degree of divergence]] of these [[point-supported distributions]] is non-negative

\begin{displaymath}
deg( \partial_\alpha \delta_0)
  =
  {\vert \alpha\vert} \geq 0  
  \,.
\end{displaymath}
This implies that $c^\alpha = 0$ for all $\alpha$, hence that the two extensions coincide.

Regarding existence:

Let

\begin{displaymath}
b \in C^\infty_{cp}(\mathbb{R}^n)
\end{displaymath}
be a [[bump function]] which is $\leq 1$ and [[constant function|constant]] on 1 over a [[neighbourhood]] of the origin. Write

\begin{displaymath}
\chi \coloneqq 1 - b \;\in\; C^\infty(\mathbb{R}^n)
\end{displaymath}
\begin{quote}%
graphics grabbed from \hyperlink{Duetsch18}{Dütsch 18, p. 108}


\end{quote}
and for $\lambda \in (0,\infty)$ a [[positive real number]], write

\begin{displaymath}
\chi_\lambda(x)
  \coloneqq
  \chi(\lambda x)
  \,.
\end{displaymath}
Since the [[product of distributions|product]] $\chi_\lambda u$ has [[support of a distribution]] on a [[complement]] of a [[neighbourhood]] of the origin, we may extend it by zero to a distribution on all of $\mathbb{R}^n$, which we will denote by the same symbols:

\begin{displaymath}
\chi_\lambda u \in \mathcal{D}'(\mathbb{R}^n)
  \,.
\end{displaymath}
By construction $\chi_\lambda u$ coincides with $u$ away from a neighbourhood of the origin, which moreover becomes arbitrarily small as $\lambda$ increases. This means that if the following [[limit of a sequence|limit]] exists

\begin{displaymath}
\hat u
  \;\coloneqq\;
  \underset{\lambda \to \infty}{\lim}
   \chi_\lambda u
\end{displaymath}
then it is an extension of $u$.

To see that the limit exists, it is sufficient to observe that we have a [[Cauchy sequence]], hence that for all $b\in C^\infty_{cp}(\mathbb{R}^n)$ the difference

\begin{displaymath}
(\chi_{n+1} u - \chi_n u)(b)
  \;=\;
  u(b)( \chi_{n+1} + \chi_n )
\end{displaymath}
becomes arbitrarily small.

It remains to see that the unique extension $\hat u$ thus established has the same scaling degree as $u$. This is shown in (\hyperlink{BrunettiFredenhagen00}{Brunetti-Fredenhagen 00, p. 24}).

\end{proof}
\begin{prop}
\label{SpaceOfPointExtensions}\hypertarget{SpaceOfPointExtensions}{}
\textbf{(space of [[point-extensions of distributions]])}

For $n \in \mathbb{N}$, let $u \in \mathcal{D}'(\mathbb{R}^n \setminus \{0\})$ be a [[distribution]] of [[scaling degree of a distribution|degree of divergence]] $deg(u) \lt \infty$.

Then $u$ does admit at least one [[extension of distributions|extension]] (def. \ref{ExtensionOfDistributions}) to a distribution $\hat u \in \mathcal{D}'(\mathbb{R}^n)$, and every choice of extension has the same [[degree of divergence of a distribution|degree of divergence]] as $u$

\begin{displaymath}
deg(\hat u) = deg(u)
  \,.
\end{displaymath}
Moreover, any two such extensions $\hat u$ and ${\hat u}^\prime$ differ by a linear combination of [[partial derivatives|partial]] [[derivatives of distributions]] of order $\leq deg(u)$ of the [[delta distribution]] $\delta_0$ [[support of a distribution|supported]] at the origin:

\begin{displaymath}
{\hat u}^\prime
  -
  \hat u
  \;=\;
  \underset{
    { \alpha \in \mathbb{N}^n }
    \atop
    { {\vert \alpha\vert} \leq deg(u) }
  }{\sum}
  q^\alpha
  \partial_\alpha \delta_0
  \,,
\end{displaymath}
for a finite number of constants $q^\alpha \in \mathbb{C}$.

\end{prop}
This is essentially (\hyperlink{Hoermander90}{Hörmander 90, thm. 3.2.4}). We follow (\hyperlink{BrunettiFredenhagen00}{Brunetti-Fredenhagen 00, theorem 5.3}), which was inspired by (\hyperlink{EpsteinGlaser73}{Epstein-Glaser 73, section 5}). Review of this approach is in (\hyperlink{Duetsch18}{Dütsch 18, theorem 3.35 (b)}), see also remark \ref{WExtensions} below.

\begin{proof}
For $f \in C^\infty(\mathbb{R}^n)$ a [[smooth function]], and $\rho \in \mathbb{N}$, we say that \emph{$f$ vanishes to order $\rho$} at the origin if all [[partial derivatives]] with multi-index $\alpha \in \mathbb{N}^n$ of total order ${\vert \alpha\vert} \leq \rho$ vanish at the origin:

\begin{displaymath}
\partial_\alpha f (0) = 0
  \phantom{AAA}
  {\vert \alpha\vert} \leq \rho
  \,.
\end{displaymath}
By [[Hadamard's lemma]], such a function may be written in the form

\begin{equation}
f(x)
  \;=\;
  \underset{
    {\alpha \in \mathbb{N}^n}
    \atop
    { {\vert \alpha \vert} = \rho + 1 }
  }{\sum}
  x^\alpha
  r_\alpha(x)
\label{ForVanishingOrderRhoHadamardExpansion}\end{equation}
for [[smooth functions]] $r_\alpha \in C^\infty_{cp}(\mathbb{R}^n)$.

Write

\begin{displaymath}
\mathcal{D}_\rho(\mathbb{R}^n)
    \hookrightarrow
  \mathcal{D}(\mathbb{R}^n)
    \coloneqq
  C^\infty_{cp}(\mathbb{R}^n)
\end{displaymath}
for the subspace of that of all [[bump functions]] on those that vanish to order $\rho$ at the origin.

By definition this is equivalently the joint [[kernel]] of the [[partial derivative|partial]] [[derivatives of distributions]] of order ${\vert \alpha\vert}$ of the [[delta distribution]] $\delta_0$ [[support of a distribution|supported]] at the origin:

\begin{displaymath}
b \in \mathcal{D}_\rho(\mathbb{R}^n)
  \phantom{AA}
  \Leftrightarrow
  \phantom{AA}
  \underset{ 
    {\alpha \in \mathbb{N}^n} 
      \atop 
    { {\vert \alpha\vert} \leq \rho }  
  }
  {\forall}  
    \left\langle \partial_\alpha \delta_0, b \right\rangle 
  = 0
  \,.
\end{displaymath}
Therefore every [[continuous linear map|continuous linear]] [[projection]]

\begin{displaymath}
p_\rho
   \;\colon\;
  \mathcal{D}(\mathbb{R}^n)
    \longrightarrow
  \mathcal{D}_\rho(\mathbb{R}^n)
\end{displaymath}
may be obtained from a choice of \emph{dual basis} to the $\{\partial_\alpha \delta_0\}$, hence a choice of smooth functions

\begin{displaymath}
\left\{
    w^\beta \in C^\infty_{cp}(\mathbb{R}^n)
  \right\}_{ { \beta \in \mathbb{N}^n } \atop { {\vert \beta\vert} \leq \rho } }
\end{displaymath}
such that

\begin{displaymath}
\left\langle
    \partial_\alpha \delta_0
    \,,\,
    w^\beta
  \right\rangle
  \;=\;
  \delta_\alpha^\beta
  \phantom{AAA}
  \Leftrightarrow
  \phantom{AAA}
  \partial_\alpha w^\beta(0)
    \;=\;
  \delta_\alpha^\beta
  \phantom{AAAA}
  \text{for}\, {\vert \alpha\vert} \leq \rho
  \,,
\end{displaymath}
by setting

\begin{equation}
p_\rho
  \;\coloneqq\;
  id
  \;-\;
  \left\langle
  \underset{
    {\alpha \in \mathbb{N}^n}
    \atop
    { {\vert \alpha\vert} \leq \rho }
  }{\sum}
  w^\alpha \partial_\alpha \delta_0
  \,,\,
  (-)
  \right\rangle
  \,,
\label{SpaceOfSmoothFunctionsOfGivenVaishingOrderProjector}\end{equation}
hence

\begin{displaymath}
p_\rho
  \;\colon\;
  b
    \mapsto
  b 
    - 
  \underset{
    {\alpha \in \mathbb{N}^n}
    \atop
    { {\vert \alpha\vert} \leq \rho }
  }{\sum}
  (-1)^{{\vert \alpha\vert}} w^\alpha \partial_\alpha b(0)
  \,.
\end{displaymath}
Together with [[Hadamard's lemma]] in the form \eqref{ForVanishingOrderRhoHadamardExpansion} this means that every $b \in \mathcal{D}(\mathbb{R}^n)$ is decomposed as

\begin{equation}
\begin{aligned}
  b(x)
  & =
  p_\rho(b)(x)
  \;+\;
  (id - p_\rho)(b)(x)
  \\
  & =
  \underset{
    { \alpha \in \mathbb{N}^n }
    \atop
    { {\vert \alpha \vert} = \rho + 1 }
  }{\sum}
  x^\alpha r_\alpha(x)
  \;+\;
  \underset{
    { \alpha \in \mathbb{N}^n }
    \atop
    { {\vert \alpha \vert} \leq \rho }
  }{\sum}
  (-1)^{{\vert \alpha \vert}}
  w^\alpha \partial_\alpha b(0)
  \end{aligned}
\label{ForExtensionOfDistributionsTestFunctionDecomposition}\end{equation}
Now let

\begin{displaymath}
\rho
  \;\coloneqq\;
  deg(u)
  \,.
\end{displaymath}
Observe that (by \href{scaling+degree+of+a+distribution#ScalingDegreeOfDistributionsBasicProperties}{this prop.}) the [[degree of divergence of a distribution|degree of divergence]] of the [[product of distributions]] $x^\alpha u$ with ${\vert \alpha\vert} = \rho + 1$ is [[negative number|negative]]

\begin{displaymath}
\begin{aligned}
    deg\left(
      x^\alpha u
    \right)
    & =
    \rho - {\vert \alpha \vert}
    \leq
    -1
 \end{aligned}
\end{displaymath}
Therefore prop. \ref{ExtensionUniqueNonPositiveDegreeOfDivergence} says that each $x^\alpha u$ for ${\vert \alpha\vert} = \rho + 1$ has a unique extension $\widehat{ x^\alpha u}$ to the origin. Accordingly the composition $u \circ p_\rho$ has a unique extension, by \eqref{ForExtensionOfDistributionsTestFunctionDecomposition}:

\begin{displaymath}
\begin{aligned}
  \left\langle
    \hat u
    \,,\,
    b
  \right\rangle
  & =
  \left\langle
    \hat u
    ,
    p_\rho(b)
  \right\rangle
  + 
  \left\langle
    \hat u
    ,
    (id - p_\rho)(b)
  \right\rangle
  \\
  & =
  \underset{
    { \alpha \in \mathbb{N}^n }
    \atop
    { {\vert \alpha \vert} = \rho + 1 }
  }{\sum}
  \underset{
    \text{unique}
  }{
  \underbrace{
    \left\langle
     \widehat{x^\alpha u}  
      \,,\,
      r_\alpha
    \right\rangle
  }
  }
  \;+\;
  \underset{
    { \alpha \in \mathbb{N}^n }
    \atop
    { {\vert \alpha\vert} \leq \rho }
  }{\sum}
  \underset{
    \text{choice}
  }{
  \underbrace{
  \langle
    \hat u
    \,,\,
    w^\alpha
  \rangle
  }
  }
  \left\langle
    \partial_\alpha \delta_0
    \,,\,
    b
  \right\rangle
  \end{aligned}
\end{displaymath}
That says that $\hat u$ is of the form

\begin{displaymath}
\hat u
  \;=\;
  \underset{
    \text{unique}
  }{
  \underbrace{
    \widehat{
      u \circ p_\rho
    }
  }
  }
  + 
  \underset{
    { \alpha \in \mathbb{N}^n  }
    \atop
    { {\vert \alpha\vert} \leq \rho }
  }{\sum}
  c^\alpha
  \,
  \partial_\alpha \delta_0
\end{displaymath}
for a finite number of constants $c^\alpha \in \mathbb{C}$.

Notice that for any extension $\hat u$ the exact value of the $c^\alpha$ here depends on the arbitrary choice of dual basis $\{w^\alpha\}$ used for this construction. But the uniqueness of the first summand means that for any two choices of extensions $\hat u$ and ${\hat u}^\prime$, their difference is of the form

\begin{displaymath}
{\hat u}^\prime - \hat u
  \;=\;
  \underset{
    { \alpha \in \mathbb{N}^n  }
    \atop
    { {\vert \alpha\vert} \leq \rho }
  }{\sum}
  ( (c')^\alpha - c^\alpha )
  \,
  \partial_\alpha \delta_0  
  \,,
\end{displaymath}
where the constants $q^\alpha \coloneqq ( (c')^\alpha - c^\alpha ) \in \mathbb{C}$ are independent of any choices.

It remains to see that all these $\hat u$ in fact have the same degree of divergence as $u$.

By \href{scaling+degree+of+a+distribution#DerivativesOfDeltaDistributionScalingDegree}{this example} the degree of divergence of the point-supported distributions on the right is $deg(\partial_\alpha \delta_0) = {\vert \alpha\vert} \leq \rho$.

Therefore to conclude it is now sufficient to show that

\begin{displaymath}
deg\left(
    \widehat{
      u \circ p_\rho
    }
  \right)
  \;=\;
  \rho
  \,.
\end{displaymath}
This is shown in (\hyperlink{BrunettiFredenhagen00}{Brunetti-Fredenhagen 00, p. 25}).

\end{proof}
\begin{remark}
\label{WExtensions}\hypertarget{WExtensions}{}
\textbf{(``W-extensions'')}

Since in \hyperlink{BrunettiFredenhagen00}{Brunetti-Fredenhagen 00, (38)} the projectors \eqref{SpaceOfSmoothFunctionsOfGivenVaishingOrderProjector} are denoted ``$W$'', the construction of [[extensions of distributions]] via the proof of prop. \ref{SpaceOfPointExtensions} has come to be called ``W-extensions'' (e.g \hyperlink{Duetsch18}{Dütsch 18}).

\end{remark}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{powers_of_feynman_propagators}{}\subsubsection*{{Powers of Feynman propagators}}\label{powers_of_feynman_propagators}

Extension of powers of [[Feynman propagators]] on [[globally hyperbolic spacetimes]] to the [[diagonal]] are worked out explicitly in (\hyperlink{Hollands07}{Hollands 07, section 3.4}).

\hypertarget{references}{}\subsection*{{References}}\label{references}

The argument for the characterization of the point extension of distributions goes back to

\begin{itemize}%
\item [[Henri Epstein]] and [[Vladimir Glaser]], section 5 of \emph{[[The role of locality in perturbation theory]]}, Annales Poincar\'e{} Phys. Theor. A 19 (1973) 211 (\href{http://www.numdam.org/item?id=AIHPA_1973__19_3_211_0}{Numdam})

\end{itemize}
thereby laying the foundation for [[causal perturbation theory]]. A textbook account in [[functional analysis]] is in

\begin{itemize}%
\item [[Lars Hörmander]], theorem 3.2.4 of \emph{The Analysis of Linear Partial Differential Operators I} (Springer, 1990, 2nd ed.)

\end{itemize}
A more concise formulation and proof is due to

\begin{itemize}%
\item [[Romeo Brunetti]], [[Klaus Fredenhagen]], section 5.2 of \emph{Microlocal analysis and interacting quantum field theories: Renormalization on Physical Backgrounds}, Commun. Math. Phys. 208 : 623-661, 2000 (\href{https://arxiv.org/abs/math-ph/9903028}{math-ph/9903028})

\end{itemize}
reviewed in

\begin{itemize}%
\item [[Michael Dütsch]], theorem 3.35 of \emph{[[From classical field theory to perturbative quantum field theory]]}, 2018

\end{itemize}
Exposition of the application to [[renormalization]] of [[Feynman diagrams]] is in

\begin{itemize}%
\item [[Christian Brouder]], \emph{Multiplication of distributions}, 2010 ([[BrouderProductOfDistributions.pdf:file]])

\end{itemize}
The refinement to the point-extension problem for distributions in the solution space of a given system of [[differential equations]] is discussed in

\begin{itemize}%
\item Dorothea Bahns, Micha Wrochna, \emph{On-shell extension of distributions}, \href{https://arxiv.org/abs/1210.5448}{arXiv:1210.5448}

\end{itemize}
Examples and applications to [[renormalization]] in [[perturbative quantum field theory]] are discussed for instance in

\begin{itemize}%
\item [[Stefan Hollands]], sections 3.3 and 3.4 of \emph{Renormalized Quantum Yang-Mills Fields in Curved Spacetime}, Rev. Math. Phys. 20:1033-1172, 2008 (\href{https://arxiv.org/abs/0705.3340}{arXiv:0705.3340})

\end{itemize}
For more on extension of distributions in [[renormalization]] see the references at \emph{[[causal perturbation theory]]} and \emph{[[locally covariant perturbative quantum field theory]]}.

[[!redirects extensions of distributions]] [[!redirects extension of a distribution]]

[[!redirects point-extension of a distribution]] [[!redirects point-extensions of distributions]]

[[!redirects point-extension of distributions]] [[!redirects point-extensions of distributions]]



\end{document}