Editing Distribution (mathematics) (section)

====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.