Editing Distribution (mathematics) (section)

===Differential operators===

====Differentiation of distributions====

Let <math>A : \mathcal{D}(U) \to \mathcal{D}(U)</math> be the partial derivative operator <math>\tfrac{\partial}{\partial x_k}.</math> To extend <math>A</math> we compute its transpose:
<math display=block>\begin{align}
\langle {}^{t}A(D_\psi), \phi \rangle
&= \int_U \psi (A\phi) \,dx && \text{(See above.)} \\
&= \int_U \psi \frac{\partial\phi}{\partial x_k} \, dx \\[4pt]
&= -\int_U \phi \frac{\partial\psi}{\partial x_k}\, dx && \text{(integration by parts)} \\[4pt]
&= -\left\langle \frac{\partial\psi}{\partial x_k}, \phi \right\rangle \\[4pt]
&= -\langle A \psi, \phi \rangle = \langle - A \psi, \phi \rangle
\end{align}</math>

Therefore <math>{}^{t}A = -A.</math> Thus, the partial derivative of <math>T</math> with respect to the coordinate <math>x_k</math> is defined by the formula
<math display=block>\left\langle \frac{\partial T}{\partial x_k}, \phi \right\rangle = - \left\langle T, \frac{\partial \phi}{\partial x_k} \right\rangle \qquad \text{ for all } \phi \in \mathcal{D}(U).</math>

With this definition, every distribution is infinitely differentiable, and the derivative in the direction <math>x_k</math> is a [[linear operator]] on <math>\mathcal{D}'(U).</math>

More generally, if <math>\alpha</math> is an arbitrary [[multi-index]], then the partial derivative <math>\partial^\alpha T</math> of the distribution <math>T \in \mathcal{D}'(U)</math> is defined by
<math display=block>\langle \partial^\alpha T, \phi \rangle = (-1)^{|\alpha|} \langle T, \partial^\alpha \phi \rangle \qquad \text{ for all } \phi \in \mathcal{D}(U).</math>

Differentiation of distributions is a continuous operator on <math>\mathcal{D}'(U);</math> this is an important and desirable property that is not shared by most other notions of differentiation.

If <math>T</math> is a distribution in <math>\R</math> then
<math display=block>\lim_{x \to 0} \frac{T - \tau_x T}{x} = T'\in \mathcal{D}'(\R),</math>
where <math>T'</math> is the derivative of <math>T</math> and <math>\tau_x</math> is a translation by <math>x;</math> thus the derivative of <math>T</math> may be viewed as a limit of quotients.{{sfn|Rudin|1991|p=180}}

====Differential operators acting on smooth functions====

A linear differential operator in <math>U</math> with smooth coefficients acts on the space of smooth functions on <math>U.</math> Given such an operator
<math display=inline>P := \sum_\alpha c_\alpha \partial^\alpha,</math>
we would like to define a continuous linear map, <math>D_P</math> that extends the action of <math>P</math> on <math>C^\infty(U)</math> to distributions on <math>U.</math> In other words, we would like to define <math>D_P</math> such that the following diagram [[Commutative diagram|commutes]]:
<math display=block>\begin{matrix}
\mathcal{D}'(U) & \stackrel{D_P}{\longrightarrow} & \mathcal{D}'(U) \\[2pt]
\uparrow & & \uparrow \\[2pt]
C^\infty(U) & \stackrel{P}{\longrightarrow} & C^\infty(U)
\end{matrix}</math>
where the vertical maps are given by assigning <math>f \in C^\infty(U)</math> its canonical distribution <math>D_f \in \mathcal{D}'(U),</math> which is defined by: 
<math display=block>D_f(\phi) = \langle f, \phi \rangle := \int_U f(x) \phi(x) \,dx \quad \text{ for all } \phi \in \mathcal{D}(U).</math> 
With this notation, the diagram commuting is equivalent to:
<math display=block>D_{P(f)} = D_PD_f \qquad \text{ for all } f \in C^\infty(U).</math>

To find <math>D_P,</math> the transpose <math>{}^{t} P : \mathcal{D}'(U) \to \mathcal{D}'(U)</math> of the continuous induced map <math>P : \mathcal{D}(U)\to \mathcal{D}(U)</math> defined by <math>\phi \mapsto P(\phi)</math> is considered in the lemma below. 
This leads to the following definition of the differential operator on <math>U</math> called {{em|the '''formal transpose''' of <math>P,</math>}} which will be denoted by <math>P_*</math> to avoid confusion with the transpose map, that is defined by
<math display=block>P_* := \sum_\alpha b_\alpha \partial^\alpha \quad \text{ where } \quad b_\alpha := \sum_{\beta \geq \alpha} (-1)^{|\beta|} \binom{\beta}{\alpha} \partial^{\beta-\alpha} c_\beta.</math>

{{math theorem|name=Lemma|math_statement= Let <math>P</math> be a linear differential operator with smooth coefficients in <math>U.</math> Then for all <math>\phi \in \mathcal{D}(U)</math> we have
<math display=block>\left\langle {}^{t}P(D_f), \phi \right\rangle = \left\langle D_{P_*(f)}, \phi \right\rangle,</math>
which is equivalent to:
<math display=block>{}^{t}P(D_f) = D_{P_*(f)}.</math>}}

{{collapse top|title=Proof|left=true}}
As discussed above, for any <math>\phi \in \mathcal{D}(U),</math> the transpose may be calculated by:
<math display=block>\begin{align}
\left\langle {}^{t}P(D_f), \phi \right\rangle &= \int_U f(x) P(\phi)(x) \,dx \\
&= \int_U f(x) \left[\sum\nolimits_\alpha c_\alpha(x) (\partial^\alpha \phi)(x) \right] \,dx \\
&= \sum\nolimits_\alpha \int_U f(x) c_\alpha(x) (\partial^\alpha \phi)(x) \,dx \\
&= \sum\nolimits_\alpha (-1)^{|\alpha|} \int_U \phi(x) (\partial^\alpha(c_\alpha f))(x) \,d x
\end{align}</math>

For the last line we used [[integration by parts]] combined with the fact that <math>\phi</math> and therefore all the functions <math>f (x)c_\alpha (x) \partial^\alpha \phi(x)</math> have compact support.<ref group="note">For example, let <math>U = \R</math> and take <math>P</math> to be the ordinary derivative for functions of one real variable and assume the support of <math>\phi</math> to be contained in the finite interval <math>(a,b),</math> then since <math>\operatorname{supp}(\phi) \subseteq (a, b)</math>
<math display=block>\begin{align}
\int_\R \phi'(x)f(x)\,dx &= \int_a^b \phi'(x)f(x) \,dx \\
&= \phi(x)f(x)\big\vert_a^b - \int_a^b f'(x) \phi(x) \,d x \\
&= \phi(b)f(b) - \phi(a)f(a) - \int_a^b f'(x) \phi(x) \,d x \\
&=-\int_a^b f'(x) \phi(x) \,d x
\end{align}</math>
where the last equality is because <math>\phi(a) = \phi(b) = 0.</math></ref> Continuing the calculation above, for all <math>\phi \in \mathcal{D}(U):</math>
<math display=block>\begin{align}
\left\langle {}^{t}P(D_f), \phi \right\rangle &=\sum\nolimits_\alpha (-1)^{|\alpha|} \int_U \phi(x) (\partial^\alpha(c_\alpha f))(x) \,dx && \text{As shown above} \\[4pt]
&= \int_U \phi(x) \sum\nolimits_\alpha (-1)^{|\alpha|} (\partial^\alpha(c_\alpha f))(x)\,dx \\[4pt]
&= \int_U \phi(x) \sum_\alpha \left[\sum_{\gamma \le \alpha} \binom{\alpha}{\gamma} (\partial^{\gamma}c_\alpha)(x) (\partial^{\alpha-\gamma}f)(x) \right] \,dx && \text{Leibniz rule}\\
&= \int_U \phi(x) \left[\sum_\alpha \sum_{\gamma \le \alpha} (-1)^{|\alpha|} \binom{\alpha}{\gamma} (\partial^{\gamma}c_\alpha)(x) (\partial^{\alpha-\gamma}f)(x)\right] \,dx \\
&= \int_U \phi(x) \left[ \sum_\alpha \left[ \sum_{\beta \geq \alpha} (-1)^{|\beta|} \binom{\beta}{\alpha} \left(\partial^{\beta-\alpha}c_{\beta}\right)(x) \right] (\partial^\alpha f)(x)\right] \,dx && \text{Grouping terms by derivatives of } f \\
&= \int_U \phi(x) \left[\sum\nolimits_\alpha b_\alpha(x) (\partial^\alpha f)(x) \right] \, dx && b_\alpha:=\sum_{\beta \geq \alpha} (-1)^{|\beta|} \binom{\beta}{\alpha} \partial^{\beta-\alpha}c_{\beta} \\
&= \left\langle \left(\sum\nolimits_\alpha b_\alpha \partial^\alpha \right) (f), \phi \right\rangle
\end{align}</math>
{{collapse bottom}}

The Lemma combined with the fact that the formal transpose of the formal transpose is the original differential operator, that is, <math>P_{**}= P,</math>{{sfn|Trèves|2006|pp=247-252}} enables us to arrive at the correct definition: the formal transpose induces the (continuous) canonical linear operator <math>P_* : C_c^\infty(U) \to C_c^\infty(U)</math> defined by <math>\phi \mapsto P_*(\phi).</math> We claim that the transpose of this map, <math>{}^{t}P_* : \mathcal{D}'(U) \to \mathcal{D}'(U),</math> can be taken as <math>D_P.</math> To see this, for every <math>\phi \in \mathcal{D}(U),</math> compute its action on a distribution of the form <math>D_f</math> with <math>f \in C^\infty(U)</math>:

<math display=block>\begin{align}
\left\langle {}^{t}P_*\left(D_f\right),\phi \right\rangle &= \left\langle D_{P_{**}(f)}, \phi \right\rangle && \text{Using Lemma above with } P_* \text{ in place of } P\\
&= \left\langle D_{P(f)}, \phi \right\rangle && P_{**} = P
\end{align}</math>

We call the continuous linear operator <math>D_P := {}^{t}P_* : \mathcal{D}'(U) \to \mathcal{D}'(U)</math> the '''{{em|differential operator on distributions extending <math>P</math>}}'''.{{sfn|Trèves|2006|pp=247-252}} Its action on an arbitrary distribution <math>S</math> is defined via:
<math display=block>D_P(S)(\phi) = S\left(P_*(\phi)\right) \quad \text{ for all } \phi \in \mathcal{D}(U).</math>

If <math>(T_i)_{i=1}^\infty</math> converges to <math>T \in \mathcal{D}'(U)</math> then for every multi-index <math>\alpha, (\partial^\alpha T_i)_{i=1}^\infty</math> converges to <math>\partial^\alpha T \in \mathcal{D}'(U).</math>

====Multiplication of distributions by smooth functions====

A differential operator of order 0 is just multiplication by a smooth function. And conversely, if <math>f</math> is a smooth function then <math>P := f(x)</math> is a differential operator of order 0, whose formal transpose is itself (that is, <math>P_* = P</math>). The induced differential operator <math>D_P : \mathcal{D}'(U) \to \mathcal{D}'(U)</math> maps a distribution <math>T</math> to a distribution denoted by <math>fT := D_P(T).</math> We have thus defined the multiplication of a distribution by a smooth function.

We now give an alternative presentation of the multiplication of a distribution <math>T</math> on <math>U</math> by a smooth function <math>m : U \to \R.</math> The product <math>mT</math> is defined by
<math display=block>\langle mT, \phi \rangle = \langle T, m\phi \rangle \qquad \text{ for all } \phi \in \mathcal{D}(U).</math>

This definition coincides with the transpose definition since if <math>M : \mathcal{D}(U) \to \mathcal{D}(U)</math> is the operator of multiplication by the function <math>m</math> (that is, <math>(M\phi)(x) = m(x)\phi(x)</math>), then
<math display=block>\int_U (M \phi)(x) \psi(x)\,dx = \int_U m(x) \phi(x) \psi(x)\,d x = \int_U \phi(x) m(x) \psi(x) \,d x = \int_U \phi(x) (M \psi)(x)\,d x,</math>
so that <math>{}^tM = M.</math>

Under multiplication by smooth functions, <math>\mathcal{D}'(U)</math> is a [[Module (mathematics)|module]] over the [[ring (mathematics)|ring]] <math>C^\infty(U).</math> With this definition of multiplication by a smooth function, the ordinary [[product rule]] of calculus remains valid. However, some unusual identities also arise. For example, if <math>\delta</math> is the Dirac delta distribution on <math>\R,</math> then <math>m \delta = m(0) \delta,</math> and if <math>\delta^'</math> is the derivative of the delta distribution, then
<math display=block>m\delta' = m(0) \delta' - m' \delta = m(0) \delta' - m'(0) \delta.</math>

The bilinear multiplication map <math>C^\infty(\R^n) \times \mathcal{D}'(\R^n) \to \mathcal{D}'\left(\R^n\right)</math> given by <math>(f,T) \mapsto fT</math> is {{em|not}} continuous; it is however, [[hypocontinuous]].{{sfn|Trèves|2006|p=423}}

'''Example.''' The product of any distribution <math>T</math> with the function that is identically {{math|1}} on <math>U</math> is equal to <math>T.</math>

'''Example.''' Suppose <math>(f_i)_{i=1}^\infty</math> is a sequence of test functions on <math>U</math> that converges to the constant function <math>1 \in C^\infty(U).</math> For any distribution <math>T</math> on <math>U,</math> the sequence <math>(f_i T)_{i=1}^\infty</math> converges to <math>T \in \mathcal{D}'(U).</math>{{sfn|Trèves|2006|p=261}}

If <math>(T_i)_{i=1}^\infty</math> converges to <math>T \in \mathcal{D}'(U)</math> and <math>(f_i)_{i=1}^\infty</math> converges to <math>f \in C^\infty(U)</math> then <math>(f_i T_i)_{i=1}^\infty</math> converges to <math>fT \in \mathcal{D}'(U).</math>

=====Problem of multiplying distributions=====

It is easy to define the product of a distribution with a smooth function, or more generally the product of two distributions whose [[singular support]]s are disjoint.<ref name="StackOverflow">{{cite web|url=https://math.stackexchange.com/q/2338283|title=Multiplication of two distributions whose singular supports are disjoint|date=Jun 27, 2017|publisher=Stack Exchange Network|author=Per Persson (username: md2perpe)}}</ref> With more effort, it is possible to define a well-behaved product of several distributions provided their [[wave front set]]s at each point are compatible. A limitation of the theory of distributions (and hyperfunctions) is that there is no associative product of two distributions extending the product of a distribution by a smooth function, as has been proved by [[Laurent Schwartz]] in the 1950s. For example, if <math>\operatorname{p.v.} \frac{1}{x}</math> is the distribution obtained by the [[Cauchy principal value]]
<math display=block>\left(\operatorname{p.v.} \frac{1}{x}\right)(\phi) = \lim_{\varepsilon\to 0^+} \int_{|x| \geq \varepsilon} \frac{\phi(x)}{x}\, dx \quad \text{ for all } \phi \in \mathcal{S}(\R).</math>

If <math>\delta</math> is the Dirac delta distribution then
<math display=block>(\delta \times x) \times \operatorname{p.v.} \frac{1}{x} = 0</math>
but,
<math display=block>\delta \times \left(x \times \operatorname{p.v.} \frac{1}{x}\right) = \delta</math>
so the product of a distribution by a smooth function (which is always well-defined) cannot be extended to an [[Associativity|associative]] product on the space of distributions.

Thus, nonlinear problems cannot be posed in general and thus are not solved within distribution theory alone. In the context of [[quantum field theory]], however, solutions can be found. In more than two spacetime dimensions the problem is related to the [[Regularization (physics)|regularization]] of [[Ultraviolet divergence|divergences]]. Here [[Henri Epstein]] and [[Vladimir Glaser]] developed the mathematically rigorous (but extremely technical) {{em|[[causal perturbation theory]]}}. This does not solve the problem in other situations. Many other interesting theories are non-linear, like for example the [[Navier–Stokes equations]] of [[fluid dynamics]].

Several not entirely satisfactory{{Citation needed|reason=Why are they not satisfactory?|date=July 2019}} theories of [[Algebra (ring theory)|algebra]]s of [[generalized function]]s have been developed, among which [[Colombeau algebra|Colombeau's (simplified) algebra]] is maybe the most popular in use today.

Inspired by Lyons' [[rough path]] theory,<ref>{{Cite journal|last1=Lyons|first1=T.|title=Differential equations driven by rough signals|doi=10.4171/RMI/240|journal=Revista Matemática Iberoamericana|pages=215–310|year=1998|volume=14 |issue=2 |doi-access=free}}</ref> [[Martin Hairer]] proposed a consistent way of multiplying distributions with certain structures ([[regularity structures]]<ref>{{cite journal|last1=Hairer|first1=Martin|title=A theory of regularity structures|journal=Inventiones Mathematicae|date=2014|doi=10.1007/s00222-014-0505-4|volume=198|issue=2|pages=269–504|bibcode=2014InMat.198..269H|arxiv=1303.5113|s2cid=119138901 }}</ref>), available in many examples from stochastic analysis, notably stochastic partial differential equations. See also Gubinelli–Imkeller–Perkowski (2015) for a related development based on [[Jean-Michel Bony|Bony]]'s [[paraproduct]] from Fourier analysis.