Editing Distribution (mathematics) (section)

===Convolution===

Under some circumstances, it is possible to define the [[convolution]] of a function with a distribution, or even the convolution of two distributions.
Recall that if <math>f</math> and <math>g</math> are functions on <math>\R^n</math> then we denote by <math>f\ast g</math> {{em|the '''convolution''' of <math>f</math> and <math>g,</math>}} defined at <math>x \in \R^n</math> to be the integral
<math display=block>(f \ast g)(x) := \int_{\R^n} f(x-y) g(y) \,dy = \int_{\R^n} f(y)g(x-y) \,dy</math>
provided that the integral exists. If <math>1 \leq p, q, r \leq \infty</math> are such that <math display=inline>\frac{1}{r} = \frac{1}{p} + \frac{1}{q} - 1</math> then for any functions <math>f \in L^p(\R^n)</math> and <math>g \in L^q(\R^n)</math> we have <math>f \ast g \in L^r(\R^n)</math> and <math>\|f\ast g\|_{L^r} \leq \|f\|_{L^p} \|g\|_{L^q}.</math>{{sfn|Trèves|2006|pp=278-283}} If <math>f</math> and <math>g</math> are continuous functions on <math>\R^n,</math> at least one of which has compact support, then <math>\operatorname{supp}(f \ast g) \subseteq \operatorname{supp} (f) + \operatorname{supp} (g)</math> and if <math>A\subseteq \R^n</math> then the value of <math>f\ast g</math> on <math>A</math> do {{em|not}} depend on the values of <math>f</math> outside of the [[Minkowski sum]] <math>A -\operatorname{supp} (g) = \{a-s : a\in A, s\in \operatorname{supp}(g)\}.</math>{{sfn|Trèves|2006|pp=278-283}}

Importantly, if <math>g \in L^1(\R^n)</math> has compact support then for any <math>0 \leq k \leq \infty,</math> the convolution map <math>f \mapsto f \ast g</math> is continuous when considered as the map <math>C^k(\R^n) \to C^k(\R^n)</math> or as the map <math>C_c^k(\R^n) \to C_c^k(\R^n).</math>{{sfn|Trèves|2006|pp=278-283}}

====Translation and symmetry====

Given <math>a \in \R^n,</math> the translation operator <math>\tau_a</math> sends <math>f : \R^n \to \Complex</math> to <math>\tau_a f : \R^n \to \Complex,</math> defined by <math>\tau_a f(y) = f(y-a).</math> This can be extended by the transpose to distributions in the following way: given a distribution <math>T,</math> {{em|the '''translation''' of <math>T</math> by <math>a</math>}} is the distribution <math>\tau_a T : \mathcal{D}(\R^n) \to \Complex</math> defined by <math>\tau_a T(\phi) := \left\langle T, \tau_{-a} \phi \right\rangle.</math>{{sfn|Trèves|2006|pp=284-297}}<ref>See for example {{harvnb|Rudin|1991|loc=§6.29}}.</ref>

Given <math>f : \R^n \to \Complex,</math> define the function <math>\tilde{f} : \R^n \to \Complex</math> by <math>\tilde{f}(x) := f(-x).</math> Given a distribution <math>T,</math> let <math>\tilde{T} : \mathcal{D}(\R^n) \to \Complex</math> be the distribution defined by <math>\tilde{T}(\phi) := T \left(\tilde{\phi}\right).</math> The operator <math>T \mapsto \tilde{T}</math> is called '''{{em|the symmetry with respect to the origin}}'''.{{sfn|Trèves|2006|pp=284-297}}

====Convolution of a test function with a distribution====

Convolution with <math>f \in \mathcal{D}(\R^n)</math> defines a linear map:
<math display=block>\begin{alignat}{4}
C_f : \,& \mathcal{D}(\R^n) && \to    \,&& \mathcal{D}(\R^n) \\
        & g                 && \mapsto\,&& f \ast g \\
\end{alignat}</math>
which is [[continuous function|continuous]] with respect to the canonical [[LF space]] topology on <math>\mathcal{D}(\R^n).</math>

Convolution of <math>f</math> with a distribution <math>T \in \mathcal{D}'(\R^n)</math> can be defined by taking the transpose of <math>C_f</math> relative to the duality pairing of <math>\mathcal{D}(\R^n)</math> with the space <math>\mathcal{D}'(\R^n)</math> of distributions.{{sfn|Trèves|2006|loc=Chapter 27}} If <math>f, g, \phi \in \mathcal{D}(\R^n),</math> then by [[Fubini's theorem]]
<math display=block>\langle C_fg, \phi \rangle = \int_{\R^n}\phi(x)\int_{\R^n}f(x-y) g(y) \,dy \,dx = \left\langle g,C_{\tilde{f}}\phi \right\rangle.</math>

Extending by continuity, the convolution of <math>f</math> with a distribution <math>T</math> is defined by
<math display=block>\langle f \ast T, \phi \rangle = \left\langle T, \tilde{f} \ast \phi \right\rangle, \quad \text{ for all } \phi \in \mathcal{D}(\R^n).</math>

An alternative way to define the convolution of a test function <math>f</math> and a distribution <math>T</math> is to use the translation operator <math>\tau_a.</math> The convolution of the compactly supported function <math>f</math> and the distribution <math>T</math> is then the function defined for each <math>x \in \R^n</math> by
<math display=block>(f \ast T)(x) = \left\langle T, \tau_x \tilde{f} \right\rangle.</math>

It can be shown that the convolution of a smooth, compactly supported function and a distribution is a smooth function. If the distribution <math>T</math> has compact support, and if <math>f</math> is a polynomial (resp. an exponential function, an analytic function, the restriction of an entire analytic function on <math>\Complex^n</math> to <math>\R^n,</math> the restriction of an entire function of exponential type in <math>\Complex^n</math> to <math>\R^n</math>), then the same is true of <math>T \ast f.</math>{{sfn|Trèves|2006|pp=284-297}} If the distribution <math>T</math> has compact support as well, then <math>f\ast T</math> is a compactly supported function, and the [[Titchmarsh convolution theorem]] {{harvtxt|Hörmander|1983|loc=Theorem 4.3.3}} implies that:
<math display=block>\operatorname{ch}(\operatorname{supp}(f \ast T)) = \operatorname{ch}(\operatorname{supp}(f)) + \operatorname{ch} (\operatorname{supp}(T))</math>
where <math>\operatorname{ch}</math> denotes the [[convex hull]] and <math>\operatorname{supp}</math> denotes the support.

====Convolution of a smooth function with a distribution====

Let <math>f \in C^\infty(\R^n)</math> and <math>T \in \mathcal{D}'(\R^n)</math> and assume that at least one of <math>f</math> and <math>T</math> has compact support. The '''{{em|convolution}}''' of <math>f</math> and <math>T,</math> denoted by <math>f \ast T</math> or by <math>T \ast f,</math> is the smooth function:{{sfn|Trèves|2006|pp=284-297}}
<math display=block>\begin{alignat}{4}
f \ast T : \,& \R^n && \to    \,&& \Complex \\
             & x    && \mapsto\,&& \left\langle T, \tau_x \tilde{f} \right\rangle \\
\end{alignat}</math>
satisfying for all <math>p \in \N^n</math>:
<math display=block>\begin{align}
&\operatorname{supp}(f \ast T) \subseteq \operatorname{supp}(f)+ \operatorname{supp}(T) \\[6pt]
&\text{ for all } p \in \N^n: \quad
\begin{cases}\partial^p \left\langle T, \tau_x \tilde{f} \right\rangle = \left\langle T, \partial^p \tau_x \tilde{f} \right\rangle \\
\partial^p (T \ast f) = (\partial^p T) \ast f = T \ast (\partial^p f).
\end{cases}
\end{align}</math>

Let <math>M</math> be the map <math>f \mapsto T \ast f</math>. If <math>T</math> is a distribution, then <math>M</math> is continuous as a map <math>\mathcal{D}(\R^n) \to C^\infty(\R^n)</math>. If <math>T</math> also has compact support, then <math>M</math> is also continuous as the map <math>C^\infty(\R^n) \to C^\infty(\R^n)</math> and continuous as the map <math>\mathcal{D}(\R^n) \to \mathcal{D}(\R^n).</math>{{sfn|Trèves|2006|pp=284-297}}

If <math>L : \mathcal{D}(\R^n) \to C^\infty(\R^n)</math> is a continuous linear map such that <math>L \partial^\alpha \phi = \partial^\alpha L \phi</math> for all <math>\alpha</math> and all <math>\phi \in \mathcal{D}(\R^n)</math> then there exists a distribution <math>T \in \mathcal{D}'(\R^n)</math> such that <math>L \phi = T \circ \phi</math> for all <math>\phi \in \mathcal{D}(\R^n).</math>{{sfn|Rudin|1991|pp=149-181}}

'''Example.'''{{sfn|Rudin|1991|pp=149-181}} Let <math>H</math> be the [[Heaviside step function|Heaviside function]] on <math>\R.</math> For any <math>\phi \in \mathcal{D}(\R),</math>
<math display=block>(H \ast \phi)(x) = \int_{-\infty}^x \phi(t) \, dt.</math>

Let <math>\delta</math> be the Dirac measure at 0 and let <math>\delta'</math> be its derivative as a distribution. Then <math>\delta' \ast H = \delta</math> and <math>1 \ast \delta' = 0.</math> Importantly, the associative law fails to hold:
<math display=block>1 = 1 \ast \delta = 1 \ast (\delta' \ast H ) \neq (1 \ast \delta') \ast H = 0 \ast H = 0.</math>

====Convolution of distributions====

It is also possible to define the convolution of two distributions <math>S</math> and <math>T</math> on <math>\R^n,</math> provided one of them has compact support. Informally, to define <math>S \ast T</math> where <math>T</math> has compact support, the idea is to extend the definition of the convolution <math>\,\ast\,</math> to a linear operation on distributions so that the associativity formula
<math display=block>S \ast (T \ast \phi) = (S \ast T) \ast \phi</math>
continues to hold for all test functions <math>\phi.</math><ref>{{harvnb|Hörmander|1983|loc=§IV.2}} proves the uniqueness of such an extension.</ref>

It is also possible to provide a more explicit characterization of the convolution of distributions.{{sfn|Trèves|2006|loc=Chapter 27}} Suppose that <math>S</math> and <math>T</math> are distributions and that <math>S</math> has compact support. Then the linear maps
<math display=block>\begin{alignat}{9}
\bullet \ast \tilde{S} : \,& \mathcal{D}(\R^n) && \to    \,&& \mathcal{D}(\R^n) && \quad \text{ and } \quad && \bullet \ast \tilde{T} : \,&& \mathcal{D}(\R^n) && \to    \,&& \mathcal{D}(\R^n) \\
    & f                       && \mapsto\,&& f \ast \tilde{S}    &&       &&     && f                       && \mapsto\,&& f \ast \tilde{T} \\
\end{alignat}</math>
are continuous. The transposes of these maps:
<math display=block>{}^{t}\left(\bullet \ast \tilde{S}\right) : \mathcal{D}'(\R^n) \to \mathcal{D}'(\R^n) \qquad {}^{t}\left(\bullet \ast \tilde{T}\right) : \mathcal{E}'(\R^n) \to \mathcal{D}'(\R^n)</math>
are consequently continuous and it can also be shown that{{sfn|Trèves|2006|pp=284-297}}
<math display=block>{}^{t}\left(\bullet \ast \tilde{S}\right)(T) = {}^{t}\left(\bullet \ast \tilde{T}\right)(S).</math>

This common value is called {{em|the '''convolution''' of <math>S</math> and <math>T</math>}} and it is a distribution that is denoted by <math>S \ast T</math> or <math>T \ast S.</math> It satisfies <math>\operatorname{supp} (S \ast T) \subseteq \operatorname{supp}(S) + \operatorname{supp}(T).</math>{{sfn|Trèves|2006|pp=284-297}} If <math>S</math> and <math>T</math> are two distributions, at least one of which has compact support, then for any <math>a \in \R^n,</math> <math>\tau_a(S \ast T) = \left(\tau_a S\right) \ast T = S \ast \left(\tau_a T\right).</math>{{sfn|Trèves|2006|pp=284-297}} If <math>T</math> is a distribution in <math>\R^n</math> and if <math>\delta</math> is a [[Dirac measure]] then <math>T \ast \delta = T = \delta \ast T</math>;{{sfn|Trèves|2006|pp=284-297}} thus <math>\delta</math> is the [[identity element]] of the convolution operation. Moreover, if <math>f</math> is a function then <math>f \ast \delta^{\prime} = f^{\prime} = \delta^{\prime} \ast f</math> where now the associativity of convolution implies that <math>f^{\prime} \ast g = g^{\prime} \ast f</math> for all functions <math>f</math> and <math>g.</math>

Suppose that it is <math>T</math> that has compact support. For <math>\phi \in \mathcal{D}(\R^n)</math> consider the function
<math display=block>\psi(x) = \langle T, \tau_{-x} \phi \rangle.</math>

It can be readily shown that this defines a smooth function of <math>x,</math> which moreover has compact support. The convolution of <math>S</math> and <math>T</math> is defined by
<math display=block>\langle S \ast T, \phi \rangle = \langle S, \psi \rangle.</math>

This generalizes the classical notion of [[convolution]] of functions and is compatible with differentiation in the following sense: for every multi-index <math>\alpha.</math>
<math display=block>\partial^\alpha(S \ast T) = (\partial^\alpha S) \ast T = S \ast (\partial^\alpha T).</math>

The convolution of a finite number of distributions, all of which (except possibly one) have compact support, is [[associative]].{{sfn|Trèves|2006|pp=284-297}}

This definition of convolution remains valid under less restrictive assumptions about <math>S</math> and <math>T.</math><ref>See for instance {{harvnb|Gel'fand|Shilov|1966–1968|loc=v. 1, pp. 103–104}} and {{harvnb|Benedetto|1997|loc=Definition 2.5.8}}.</ref>

The convolution of distributions with compact support induces a continuous bilinear map <math>\mathcal{E}' \times \mathcal{E}' \to \mathcal{E}'</math> defined by <math>(S,T) \mapsto S * T,</math> where <math>\mathcal{E}'</math> denotes the space of distributions with compact support.{{sfn|Trèves|2006|p=423}} However, the convolution map as a function <math>\mathcal{E}' \times \mathcal{D}' \to \mathcal{D}'</math> is {{em|not}} continuous{{sfn|Trèves|2006|p=423}} although it is separately continuous.{{sfn|Trèves|2006|p=294}} The convolution maps <math>\mathcal{D}(\R^n) \times \mathcal{D}' \to \mathcal{D}'</math> and <math>\mathcal{D}(\R^n) \times \mathcal{D}' \to \mathcal{D}(\R^n)</math> given by <math>(f, T) \mapsto f * T</math> both {{em|fail}} to be continuous.{{sfn|Trèves|2006|p=423}} Each of these non-continuous maps is, however, [[separately continuous]] and [[hypocontinuous]].{{sfn|Trèves|2006|p=423}}

====Convolution versus multiplication====

In general, [[Regularization (physics)|regularity]] is required for multiplication products, and [[Principle of locality|locality]] is required for convolution products. It is expressed in the following extension of the [[Convolution theorem|Convolution Theorem]] which guarantees the existence of both convolution and multiplication products. Let <math>F(\alpha) = f \in \mathcal{O}'_C</math> be a rapidly decreasing tempered distribution or, equivalently, <math>F(f) = \alpha \in \mathcal{O}_M</math> be an ordinary (slowly growing, smooth) function within the space of tempered distributions and let <math>F</math> be the normalized (unitary, ordinary frequency) [[Fourier transform]].<ref>{{cite book|last=Folland|first=G.B.|title=Harmonic Analysis in Phase Space|publisher=Princeton University Press|publication-place=Princeton, NJ|year=1989}}</ref> Then, according to {{harvtxt|Schwartz|1951}},
<math display=block>F(f * g) = F(f) \cdot F(g) \qquad \text{ and } \qquad F(\alpha \cdot g) = F(\alpha) * F(g)</math>
hold within the space of tempered distributions.<ref>{{cite book|last=Horváth|first=John|author-link = John Horvath (mathematician)|title=Topological Vector Spaces and Distributions|publisher=Addison-Wesley Publishing Company|publication-place=Reading, MA|year=1966}}</ref><ref>{{cite book|last=Barros-Neto|first=José|title=An Introduction to the Theory of Distributions|publisher=Dekker|publication-place=New York, NY|year=1973}}</ref><ref>{{cite book|last=Petersen|first=Bent E.|title=Introduction to the Fourier Transform and Pseudo-Differential Operators|publisher=Pitman Publishing|publication-place=Boston, MA|year=1983}}</ref> In particular, these equations become the [[Poisson summation formula|Poisson Summation Formula]] if <math>g \equiv \operatorname{\text{Ш}}</math> is the [[Dirac comb|Dirac Comb]].<ref>{{cite book|last=Woodward|first=P.M.|title=Probability and Information Theory with Applications to Radar|publisher=Pergamon Press|publication-place=Oxford, UK|year=1953}}</ref> The space of all rapidly decreasing tempered distributions is also called the space of {{em|convolution operators}} <math>\mathcal{O}'_C</math> and the space of all ordinary functions within the space of tempered distributions is also called the space of {{em|multiplication operators}} <math>\mathcal{O}_M.</math> More generally, <math>F(\mathcal{O}'_C) = \mathcal{O}_M</math> and <math>F(\mathcal{O}_M) = \mathcal{O}'_C.</math>{{sfn|Trèves|2006|pp=318-319}}<ref>{{cite book|last1=Friedlander|first1=F.G.|last2=Joshi|first2=M.S.|title=Introduction to the Theory of Distributions|publisher=Cambridge University Press|publication-place=Cambridge, UK|year=1998}}</ref> A particular case is the [[Paley–Wiener theorem#Schwartz's Paley–Wiener theorem|Paley-Wiener-Schwartz Theorem]] which states that <math>F(\mathcal{E}') = \operatorname{PW}</math> and <math>F(\operatorname{PW} ) = \mathcal{E}'.</math> This is because <math>\mathcal{E}' \subseteq \mathcal{O}'_C</math> and <math>\operatorname{PW} \subseteq \mathcal{O}_M.</math> In other words, compactly supported tempered distributions <math>\mathcal{E}'</math> belong to the space of {{em|convolution operators}} <math>\mathcal{O}'_C</math> and
Paley-Wiener functions <math>\operatorname{PW},</math> better known as [[Bandlimiting|bandlimited functions]], belong to the space of {{em|multiplication operators}} <math>\mathcal{O}_M.</math>{{sfn|Schwartz|1951}}

For example, let <math>g \equiv \operatorname{\text{Ш}} \in \mathcal{S}'</math> be the Dirac comb and <math>f \equiv \delta \in \mathcal{E}'</math> be the [[Dirac delta function|Dirac delta]];then <math>\alpha \equiv 1 \in \operatorname{PW}</math> is the function that is constantly one and both equations yield the [[Dirac comb#Dirac-comb identity|Dirac-comb identity]]. Another example is to let <math>g</math> be the Dirac comb and <math>f \equiv \operatorname{rect} \in \mathcal{E}'</math> be the [[rectangular function]]; then <math>\alpha \equiv \operatorname{sinc} \in \operatorname{PW}</math> is the [[sinc function]] and both equations yield the [[Nyquist–Shannon sampling theorem|Classical Sampling Theorem]] for suitable <math>\operatorname{rect}</math> functions. More generally, if <math>g</math> is the Dirac comb and <math>f \in \mathcal{S} \subseteq \mathcal{O}'_C \cap \mathcal{O}_M</math> is a [[Smoothness|smooth]] [[window function]] ([[Schwartz space|Schwartz function]]), for example, the [[Gaussian function|Gaussian]], then <math>\alpha \in \mathcal{S}</math> is another smooth window function (Schwartz function). They are known as [[mollifier]]s, especially in [[partial differential equation]]s theory, or as [[Regularization (mathematics)|regularizers]] in [[Regularization (physics)|physics]] because they allow turning [[generalized function]]s into [[Function (mathematics)|regular functions]].