# Distribution (mathematics)

Distributions (or generalized functions) are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta function.

The practical use of distributions can be traced back to the use of Green functions in the 1830s to solve ordinary differential equations, but was not formalized until much later. According to Kolmogorov & Fomin (1957), generalized functions originated in the work of Sergei Sobolev (1936) on second-order hyperbolic partial differential equations, and the ideas were developed in somewhat extended form by Laurent Schwartz in the late 1940s. According to his autobiography, Schwartz introduced the term "distribution" by analogy with a distribution of electrical charge, possibly including not only point charges but also dipoles and so on. Gårding (1997) comments that although the ideas in the transformative book by Schwartz (1951) were not entirely new, it was Schwartz's broad attack and conviction that distributions would be useful almost everywhere in analysis that made the difference.

Distribution theory reinterprets functions as linear functionals acting on a space of test functions. Standard functions act by integration against a test function, but many other linear functionals do not arise in this way, and these are the "generalized functions". There are different possible choices for the space of test functions, leading to different spaces of distributions. The basic space of test function consists of smooth functions with compact support, leading to standard distributions. Use of the space of smooth, rapidly (faster than any polynomial increases) decreasing test functions (these functions are called Schwartz functions) gives instead the tempered distributions, which are important because they have a well-defined distributional Fourier transform. Every tempered distribution is a distribution in the normal sense, but the converse is not true: in general the larger the space of test functions, the more restrictive the notion of distribution. On the other hand, the use of spaces of analytic test functions leads to Sato's theory of hyperfunctions; this theory has a different character from the previous ones because there are no analytic functions with non-empty compact support.

## Basic idea

A typical test function, the bump function Ψ(x). It is smooth (infinitely differentiable) and has compact support (is zero outside an interval, in this case the interval [−1, 1]).

Distributions are a class of linear functionals that map a set of test functions (conventional and well-behaved functions) into the set of real numbers. In the simplest case, the set of test functions considered is D(R), which is the set of functions $\varphi$  : RR having two properties:

• $\varphi$  is smooth (infinitely differentiable);
• $\varphi$  has compact support (is identically zero outside some bounded interval).

A distribution T is a linear mapping T : D(R) → R. Instead of writing T($\varphi$ ), it is conventional to write $\langle T,\varphi \rangle$  for the value of T acting on a test function $\varphi$ . A simple example of a distribution is the Dirac delta δ, defined by

$\left\langle \delta ,\varphi \right\rangle =\varphi (0),$

meaning that δ evaluates a test function at 0. Its physical interpretation is as the density of a point source.

As described next, there are straightforward mappings from both locally integrable functions and Radon measures to corresponding distributions, but not all distributions can be formed in this manner.

### Functions and measures as distributions

Suppose that f : RR is a locally integrable function. Then a corresponding distribution Tf may be defined by

$\left\langle T_{f},\varphi \right\rangle =\int _{\mathbf {R} }f(x)\varphi (x)\,dx\quad {\text{for}}\quad \varphi \in D(\mathbf {R} ).$

This integral is a real number which depends linearly and continuously on $\varphi$ . Conversely, the values of the distribution Tf on test functions in D(R) determine the pointwise almost everywhere values of the function f on R. In a conventional abuse of notation, f is often used to represent both the original function f and the corresponding distribution Tf. This example suggests the definition of a distribution as a linear and, in an appropriate sense, continuous functional on the space of test functions D(R).

Similarly, if μ is a Radon measure on R, then a corresponding distribution Rμ may be defined by

$\left\langle R_{\mu },\varphi \right\rangle =\int _{\mathbf {R} }\varphi \,d\mu \quad {\text{for}}\quad \varphi \in D(\mathbf {R} ).$

This integral also depends linearly and continuously on $\varphi$ , so that Rμ is a distribution. If μ is absolutely continuous with respect to Lebesgue measure with density f and dμ = f dx, then this definition for Rμ is the same as the previous one for Tf, but if μ is not absolutely continuous, then Rμ is a distribution that is not associated with a function. For example, if P is the point-mass measure on R that assigns measure one to the singleton set {0} and measure zero to sets that do not contain zero, then

$\int _{\mathbf {R} }\varphi \,dP=\varphi (0),$

so that RP = δ is the Dirac delta.

Distributions may be multiplied by real numbers and added together, so they form a real vector space. A distribution may also be multiplied by a rapidly decreasing infinitely differentiable function to get another distribution, but it is not possible to define a product of general distributions that extends the usual pointwise product of functions and has the same algebraic properties. This result was shown by Schwartz (1954), and is usually referred to as the Schwartz Impossibility Theorem.

### Derivatives of distributions

It is desirable to choose a definition for the derivative of a distribution which, at least for distributions derived from smooth functions, has the property that $T'_{f}=T_{f'}$ . If $\varphi$  is a test function, we can use integration by parts to see that

$\left\langle f',\varphi \right\rangle =\int _{\mathbf {R} }f'\varphi \,dx={\Big [}f(x)\varphi (x){\Big ]}_{-\infty }^{\infty }-\int _{\mathbf {R} }f\varphi '\,dx=-\left\langle f,\varphi '\right\rangle$

where the last equality follows from the fact that $\varphi$  has compact support, so is zero outside of a bounded set. This suggests that if $T$  is a distribution, we should define its derivative $T'$  by

$\left\langle T',\varphi \right\rangle =-\left\langle T,\varphi '\right\rangle .$

It turns out that this is the proper definition; it extends the ordinary definition of derivative, every distribution becomes infinitely differentiable and the usual properties of derivatives hold.

Example: Recall that the Dirac delta (so-called Dirac delta function) is the distribution defined by the equation

$\left\langle \delta ,\varphi \right\rangle =\varphi (0).$

It is the derivative of the distribution corresponding to the Heaviside step function H: For any test function $\varphi$ ,

$\left\langle H',\varphi \right\rangle =-\int _{-\infty }^{\infty }H(x)\varphi '(x)\,dx=-\varphi (\infty )+\varphi (0)=\left\langle \delta ,\varphi \right\rangle ,$

so H′ = δ. Note, $\varphi$ (∞) = 0 because $\varphi$  has compact support by our definition of a test function. Similarly, the derivative of the Dirac delta is the distribution defined by the equation

$\langle \delta ',\varphi \rangle =-\varphi '(0).$

This latter distribution is an example of a distribution that is not derived from a function or a measure. Its physical interpretation is the density of a dipole source. Just as the Dirac impulse can be realized in the weak limit as a sequence of various kinds of constant norm bump functions of ever increasing amplitude and narrowing support, its derivative can by definition be realized as the weak limit of the negative derivatives of said functions, which are now antisymmetric about the eventual distribution's point of singular support.

## Test functions and distributions

Throughout, n is a fixed positive integer, U is a fixed non-empty open subset of Euclidean space $\mathbb {R} ^{n}$ , and $\mathbb {N} :=\left\{0,1,\ldots \right\}$ . In the following, real-valued distributions on an open subset U will be formally defined. With minor modifications, one can also define complex-valued distributions, and one can replace Rn by any (paracompact) smooth manifold.

For any $k=0,1,\ldots ,\infty$  let $C^{k}\left(U\right)$  denote the vector space of all k-times continuously differentiable complex-valued functions on $U$ . Recall that the support of a function f, denoted by $\operatorname {supp} (f)$ , is the closure in $\operatorname {Dom} f$  of the set $\left\{x\in \operatorname {Dom} f:f(x)\neq 0\right\}$ . For any compact subset $K\subseteq U$ , let $C^{k}\left(K\right)$  denote all those functions $f\in C^{k}\left(U\right)$  such that $\operatorname {supp} (f)\subseteq K$  (note that $C^{k}\left(K\right)$  depends on both K and $U$  but we will only indicate K, where in particular, if $f\in C^{k}\left(K\right)$  then the domain of f is $U$  rather than K). Note that if f is a real-valued function on U, then f is an element of $C_{c}^{k}\left(U\right)$  if and only if it is a $C^{k}$  bump function.

Distributions on U are defined to be the continuous linear functionals on the space $C_{c}^{\infty }\left(U\right)$  of test functions on U when this space is endowed with a particular topology (called the canonical LF topology). The space of test functions on U is also denoted is also D(U) or ${\mathcal {D}}(U)$ . So to define the space of distributions, which is denoted by ${\mathcal {D}}^{\prime }\left(U\right)$ , we must first define the topology on the space $C_{c}^{\infty }\left(U\right)$ , which requires that several other topological vector spaces (TVSs) be defined first.

To define the various special subspaces of ${\mathcal {D}}^{\prime }\left(U\right)$  (such as distributions of order $\leq$  some integer, etc.), we construct the spaces $C^{k}\left(U\right)$  and $C_{c}^{k}\left(U\right)$  for arbitrary $k\in \{0,1,\ldots ,\infty \}$ , which adds almost no additional complexity to the presentation and allows us to easily define additional spaces of distributions later. Once the topology on $C_{c}^{\infty }\left(U\right)$  is defined, we can then immediately place a topology on the space of distributions.

Throughout, we will let $\mathbb {K}$  be any collection of compact subsets of $U$  such that (1) $U=\cup _{K\in \mathbb {K} }K$ , and (2) for any compact $K\subseteq U$  there exists some $K_{2}\in \mathbb {K}$  such that $K\subseteq K_{2}$ . The most common choices for $\mathbb {K}$  are:

• the set of all compact subsets of $U$ , or
• a set $\left\{{\overline {U_{1}}},{\overline {U_{2}}},\ldots \right\}$  where $U=\cup _{i=1}^{\infty }U_{i}$ , and for all i, ${\overline {U_{i}}}\subseteq U_{i+1}$  and $U_{i}$  is a relatively compact non-empty open subset of $U$  (i.e. "relatively compact" means that the closure of $U_{i}$ , in either U or $\mathbb {R} ^{n}$ , is compact).

We make $\mathbb {K}$  into a directed set by defining $K_{1}\leq K_{2}$  if and only if $K_{1}\subseteq K_{2}$ . Note that although the definitions of the subsequently defined topologies explicitly reference $\mathbb {K}$ , in reality they do not depend on the choice of $\mathbb {K}$ ; that is, if $\mathbb {K} _{1}$  and $\mathbb {K} _{2}$  are any two such collections of compact subsets of $U$ , then the topologies defined on $C^{k}\left(U\right)$  and $C_{c}^{k}\left(U\right)$  by using $\mathbb {K} _{1}$  in place of $\mathbb {K}$  are the same as those defined by using $\mathbb {K} _{2}$  in place of $\mathbb {K}$ .

### Topology on the space of Ck maps

We give $C^{k}\left(U\right)$  the Fréchet space topology defined by the family of seminorms

$p_{i,K}(f):=\sup _{|p|\leq i}\left(\sup _{x_{0}\in K}\left|{\frac {\partial ^{p}f}{\partial x^{p}}}{\bigg \vert }_{x=x_{0}}\right|\right)$

as i ranges over $0,\ldots ,k$  and K ranges over $\mathbb {K}$ , where $p:=\left(p_{1},\ldots ,p_{n}\right)$  is a multi-index of non-negative integers and $|p|:=p_{1}+\cdots +p_{n}$  denotes its length. Under this topology, a net of functions $\left(f_{\alpha }\right)$  in $C^{k}\left(U\right)$  converges to a function $f\in C^{k}\left(U\right)$  if and only if for every multi-index p with $|p|  and every $K\in \mathbb {K}$ , $\left({\frac {\partial ^{p}f_{\alpha }}{\partial x^{p}}}\right)$  converges to ${\frac {\partial ^{p}f}{\partial x^{p}}}$  uniformly on K. Note that $C^{k}\left(U\right)$  is a Montel space if and only if $k=\infty$ . 

### Topology on the space of Ck maps with support in a compact set

For any compact $K\subseteq U$ , $C^{k}\left(K\right)$  is a closed subspace of $C^{k}\left(U\right)$  and we give $C^{k}\left(K\right)$  the subspace topology induced by $C^{k}\left(U\right)$ , thereby making it into a Fréchet space. For all compact $K_{1},K_{2}\subseteq U$  with $K_{1}\subseteq K_{2}$ , denote the natural inclusion by $\operatorname {In} _{K_{1}}^{K_{2}}:C^{k}\left(K_{1}\right)\to C^{k}\left(K_{2}\right)$  where note that this map is a linear embedding of TVSs whose range is closed in its codomain (said differently, the topology on $C^{k}\left(K_{1}\right)$  is identical to the subspace topology it inherits from $C^{k}\left(K_{2}\right)$ , and $C^{k}\left(K_{1}\right)$  is a closed subspace of $C^{k}\left(K_{2}\right)$ ). When $k\neq \infty$  then $C^{k}\left(K\right)$  is a Banach space and when $k=2$ , it is even a Hilbert space. If K is non-trivial then $C^{\infty }\left(K\right)$  is not a Banach space but $C^{k}\left(K\right)$  is a Fréchet space for all $k=0,1,\ldots ,\infty$ .

### Topology on the space of test functions

Let $C_{c}^{k}\left(U\right)$  denote all those functions in $C^{k}\left(U\right)$  that have compact support[disambiguation needed] in $U$ , where note that $C_{c}^{k}\left(U\right)$  is the union of all $C^{k}\left(K\right)$  as K ranges over $\mathbb {K}$ . Moreover, for every k, $C_{c}^{k}\left(U\right)$  is a dense subset of $C^{k}\left(U\right)$ . If $k=\infty$  then $C_{c}^{\infty }\left(U\right)$  is called the space of test functions and it may also be denoted by ${\mathcal {D}}\left(U\right)$ . For all compact $K_{1},K_{2}\subseteq U$  with $K_{1}\subseteq K_{2}$ , there are natural inclusions $\operatorname {In} _{K_{1}}^{K_{2}}:C^{k}\left(K_{1}\right)\to C^{k}\left(K_{2}\right)$ , which form a direct system in the category of locally convex TVSs that is directed by $\mathbb {K}$  (under subset inclusion). This system's direct limit (in the category of locally convex TVSs) is the locally convex TVS $C_{c}^{k}\left(U\right)$  together with $\left(\operatorname {In} _{K}\right)_{K\in \mathbb {K} }$ , where $\operatorname {In} _{K}:C^{k}\left(K\right)\to C_{c}^{k}\left(U\right)$  is the natural inclusion and where the topology on $C_{c}^{k}\left(U\right)$  is the strongest locally convex topology making all of these inclusion maps continuous. This topology, called the canonical LF topology, makes $C_{c}^{k}\left(U\right)$  into a complete Hausdorff locally convex LF-space (and also an LB-space when $k\neq \infty$ ) and this topology is finer than the subspace topology that $C_{c}^{k}\left(U\right)$  inherits from $C^{k}\left(U\right)$  (thus the natural inclusion $C_{c}^{k}\left(U\right)\to C^{k}\left(U\right)$  is continuous but in general not a topological embedding). From basic category theory, we know that this topology is independent of the particular choice of the directed collection of compact subsets $\mathbb {K}$  (as mentioned earlier). The spaces $C_{c}^{\infty }\left(U\right)$ , $C^{\infty }\left(U\right)$ , and the strong duals ${\mathcal {D}}^{\prime }\left(U\right):=\left(C_{c}^{\infty }\left(U\right)\right)_{b}^{\prime }$  and ${\mathcal {E}}^{\prime }\left(U\right):=\left(C^{\infty }\left(U\right)\right)_{b}^{\prime }$  are also barreled nuclear Montel bornological Mackey spaces.

It may be shown that for any compact subset $K\subseteq U$ , the natural inclusion $\operatorname {In} _{K}:C^{k}\left(K\right)\to C_{c}^{k}\left(U\right)$  is an embedding of TVSs. Furthermore, a sequence $\left(x_{i}\right)_{i=1}^{\infty }$  in $C_{c}^{k}\left(U\right)$  converges in $C_{c}^{k}\left(U\right)$  if and only if there exists some $K\in \mathbb {K}$  such that $C^{k}\left(K\right)$  contains this sequence and this sequence converges in $C^{k}\left(K\right)$ . From the universal property of direct limits, we know that if $u:C_{c}^{k}\left(U\right)\to Y$  is a linear map into a locally convex space Y (not necessarily Hausdorff), then u is continuous $\iff$  u is bounded $\iff$  for every $K\in \mathbb {K}$ , $u$ 's restriction to $C^{k}\left(K\right)$ , $u\vert _{C^{k}\left(K\right)}:C^{k}\left(K\right)\to Y$ , is continuous (or bounded). A subset B of $C_{c}^{k}\left(U\right)$  is bounded in $C_{c}^{k}\left(U\right)$  if and only if there exists some $K\in \mathbb {K}$  such that $B\subseteq C^{k}\left(K\right)$  and B is a bounded subset of $C^{k}\left(K\right)$ . Moreover, if $K\subseteq U$  is compact and $S\subseteq C^{k}\left(K\right)$  then S is bounded in $C^{k}\left(K\right)$  if and only if it is bounded in $C^{k}\left(U\right)$ . For any $0\leq k\leq \infty$ , any bounded subset of $C_{c}^{k+1}\left(U\right)$  (resp. $C^{k+1}\left(U\right)$ ) is a relatively compact subset of $C_{c}^{k}\left(U\right)$  (resp. $C^{k}\left(U\right)$ ), where $\infty +1=\infty$ .

For all compact KU, the interior of $C^{k}\left(K\right)$  in $C_{c}^{k}\left(U\right)$  is empty so $C_{c}^{k}\left(U\right)$  is of the first category in itself. By Baire's theorem, $C_{c}^{k}\left(U\right)$  is not metrizable.

The bilinear multiplication map $C^{\infty }\left(\mathbb {R} ^{m}\right)\times C_{c}^{\infty }\left(\mathbb {R} ^{n}\right)\to C_{c}^{\infty }\left(\mathbb {R} ^{m+n}\right)$  given by $\left(f,g\right)\mapsto fg$  is not continuous; it is however, hypocontinuous.

#### Topology defined via neighborhoods

The canonical LF topology may also be defined by defining the neighborhoods of the origin as follows: if U is a convex subset of $C_{c}^{k}\left(U\right)$ , then U is a neighborhood of the origin in the canonical LF topology if and only if for all compact $K\in \mathbb {K}$ , $U\cap C^{k}\left(K\right)$  is a neighborhood of the origin in $C^{k}\left(K\right)$ .

#### Topology defined via differential operators

A linear differential operator in U with C coefficients is a sum $P\left(x,\partial /\partial x\right):=\sum _{\alpha \in \mathbb {N} ^{n}}c_{\alpha }(x)\left(\partial /\partial x\right)^{\alpha }$  where all but finitely many of the C(U) functions $c_{\alpha }$  are identically 0. The integer $\sup \left\{\left|\alpha \right|:c_{\alpha }\neq 0\right\}$  is called the order of the differential operator $P\left(x,\partial /\partial x\right)$ , where $\left|\alpha \right|:=\alpha _{1}+\cdots +\alpha _{n}$  when $\alpha =\left(\alpha _{1},\ldots ,\alpha _{n}\right)$ . If $P\left(x,\partial /\partial x\right)$  is a linear differential operator of order k then it induces a canonical linear map $C^{k}\left(U\right)\to C^{0}\left(U\right)$  defined by $\phi \mapsto P\left(x,\partial /\partial x\right)\phi$ , where we shall reuse notation and also denote this map by $P\left(x,\partial /\partial x\right)$ .

For any $1\leq k\leq \infty$ , the canonical LF topology on $C_{c}^{k}\left(U\right)$  is the weakest locally convex TVS topology making all linear differential operators in U of order < k + 1 into continuous maps from $C_{c}^{k}\left(U\right)$  into $C_{c}^{0}\left(U\right)$ .

#### Sequentional definition of the topology on D(U)

The space D(U) of test functions on U, which is a real vector space, can be given a topology by defining the limit of a sequence of elements of D(U). A sequence ($\varphi$ k) in D(U) is said to converge to $\varphi$  ∈ D(U) if the following two conditions hold:

• There is a compact set K ⊂ U containing the supports of all $\varphi$ k:
$\bigcup \nolimits _{k}\operatorname {supp} (\varphi _{k})\subset K.$
• For each multi-index α, the sequence of partial derivatives $\partial ^{\alpha }\varphi _{k}$  tends uniformly to $\partial ^{\alpha }\varphi$ .

With this definition, D(U) becomes a complete locally convex topological vector space satisfying the Heine–Borel property.

This topology can be placed in the context of the following general construction: let

$X=\bigcup \nolimits _{i}X_{i}$

be a countable increasing union of locally convex topological vector spaces and ιi : XiX be the inclusion maps. In this context, the inductive limit topology, or final topology, τ on X is the finest locally convex vector space topology making all the inclusion maps $\iota _{i}$  continuous. The topology τ can be explicitly described as follows: let β be the collection of convex balanced subsets W of X such that WXi is open for all i. A base for the inductive limit topology τ then consists of the sets of the form x + W, where x in X and W in β.

The proof that τ is a vector space topology makes use of the assumption that each Xi is locally convex. By construction, β is a local base for τ. That any locally convex vector space topology on X must necessarily contain τ means it is the weakest one. One can also show that, for each i, the subspace topology Xi inherits from τ coincides with its original topology. When each Xi is a Fréchet space, (X, τ) is called an LF space.

Now let U be the union of Ui where {Ui} is a countable nested family of open subsets of U with compact closures Ki = Ui. Then we have the countable increasing union

$\mathrm {D} (U)=\bigcup \nolimits _{i}C^{\infty }\left(K_{i}\right)$

where $C^{\infty }\left(K_{i}\right)$  is the set of all smooth functions on U with support lying in Ki. On each $C^{\infty }\left(K_{i}\right)$ , consider the topology given by the seminorms

$\|\varphi \|_{\alpha }=\max _{x\in K_{i}}\left|\partial ^{\alpha }\varphi \right|,$

i.e. the topology of uniform convergence of derivatives of arbitrary order. This makes each $C^{\infty }\left(K_{i}\right)$  a Fréchet space. The resulting LF space structure on D(U) is the topology described in the beginning of the section.

On D(U), one can also consider the topology given by the seminorms

$\|\varphi \|_{\alpha ,K_{i}}=\max _{x\in K_{i}}\left|\partial ^{\alpha }\varphi \right|.$

However, this topology has the disadvantage of not being complete. On the other hand, because of the particular features of the $C^{\infty }\left(K_{i}\right)$ 's, a set is bounded with respect to τ if and only if it lies in some $C^{\infty }\left(K_{i}\right)$ 's. The completeness of (D(U), τ) then follow from that of DKi's.

The topology τ is not metrizable by the Baire category theorem, since D(U) is the union of subspaces of the first category in D(U).

### Distributions

A distribution is a continuous linear functional on $C_{c}^{\infty }\left(U\right)$ . If T is a linear functional on $C_{c}^{\infty }\left(U\right)$  then the following are equivalent:

• T is a distribution;
• T is continuous[disambiguation needed];
• T is continuous[disambiguation needed] at the origin;
• T is bounded;
• T is sequentially continuous; i.e. for every sequence $f_{1},f_{2},\ldots$  of test functions in $C_{c}^{\infty }\left(U\right)$  that converges to f in $C_{c}^{\infty }\left(U\right)$ , $\lim _{k\to \infty }T\left(f_{i}\right)=T\left(\lim _{k\to \infty }f_{k}\right)$ ;
• Even though the topology of $C_{c}^{\infty }\left(U\right)$  is not metrizable, a linear functional on $C_{c}^{\infty }\left(U\right)$  is continuous if and only if it is sequentially continuous.
• T is sequentially continuous at the origin; i.e. for every sequence $f_{1},f_{2},\ldots$  of test functions in $C_{c}^{\infty }\left(U\right)$  that converges to 0 in $C_{c}^{\infty }\left(U\right)$ , $\lim _{k\to \infty }T\left(f_{i}\right)=0$ ;
• the kernel of T is a closed subspace of $C_{c}^{\infty }\left(U\right)$ ;
• to every compact subset K of U, there is an integer $m\geq 0$  and a constant $C>0$  such that for all $f\in C^{\infty }\left(K\right)$ ,
$\left|T(f)\right|\leq C\sup _{|p|\leq m}\left(\sup _{x\in U}\left|{\frac {\partial ^{p}f}{\partial x^{p}}}(x)\right|\right)$ ;
• the statement above but with the compact set K restricted to lie in $\mathbb {K}$ ;
• for every compact subset K of U there exists a positive constant CK and a non-negative integer NK such that
$|T(f)|\leq C_{K}\sup\{|\partial ^{\alpha }f(x)|\mid x\in K,|\alpha |\leq N_{K}\}$

for all test functions f with support contained in K.

• Note that if N can be chosen to be independent of K then the distribution is said to be of finite order and the smallest such N is called the order of the distribution. A distribution is said to have infinite order if it does not have finite order.
• the statement above but with the compact set K restricted to lie in $\mathbb {K}$ ;
• for any compact subset K of U and any sequence of test functions $f_{1},f_{2},\ldots$  belonging to $C^{\infty }\left(K\right)$ , if $\left({\frac {\partial ^{p}f}{\partial x^{p}}}\right)_{i=1}^{\infty }$  converges uniformly to zero for all multi-indices p, then $T\left(f_{i}\right)\to 0$ ;
• the statement above but with the compact set K restricted to lie in $\mathbb {K}$ ;

We have the canonical duality pairing between a distribution $T\in {\mathcal {D}}'\left(U\right)$  and a test function $f\in C_{c}^{\infty }\left(U\right)$ ), which is denoted using angle brackets by

${\begin{cases}{\mathcal {D}}'(U)\times C_{c}^{\infty }\left(U\right)\to \mathbb {R} \\(T,f)\mapsto \langle T,f\rangle :=T(f),\end{cases}}$

so that ⟨T,$f$ ⟩ = T($f$ ). One interprets this notation as the distribution T acting on the test function $f$  to give a scalar, or symmetrically as the test function $f$  acting on the distribution T.

### Topology on the space of distributions

The space of distributions, denoted by ${\mathcal {D}}^{\prime }\left(U\right)$ , is the continuous dual space of $C_{c}^{\infty }\left(U\right)$  with the topology of uniform convergence on bounded subsets of $C_{c}^{\infty }\left(U\right)$  (this is written as ${\mathcal {D}}^{\prime }\left(U\right)=\left(C_{c}^{\infty }\left(U\right)\right)_{b}^{\prime }$ ), where this topology is also called the strong dual topology (in functional analysis, the strong dual topology is the "standard" or "default" topology placed on the continuous dual space $X^{\prime }$ , where if X is a normed space then this strong dual topology is the same as the usual norm-induced topology on $X^{\prime }$ ). This topology is chosen because it is with this topology that D′(U) becomes a nuclear Montel space and it is with this topology that the kernels theorem of Schwartz holds. No matter what dual topology is placed on D′(U), a sequence of distributions converges in this topology if and only if it converges pointwise (although this need not be true of a net), which is why the topology is sometimes defined to be the weak-* topology. No matter which topology is chosen, D′(U) will be a non-metrizable, locally convex topological vector space.

Each of $C_{c}^{\infty }\left(U\right)$ , ${\mathcal {D}}^{\prime }\left(U\right)$ , $C^{\infty }\left(U\right)$ , and $\left(C^{\infty }\left(U\right)\right)_{b}^{\prime }$  are nuclear Montel spaces. One reason for giving $C_{c}^{\infty }\left(U\right)$  the canonical LF topology is because it is with this topology that $C_{c}^{\infty }\left(U\right)$  and its continuous dual space both become nuclear spaces, which have many nice properties and which may be viewed as a generalization of finite-dimensional spaces (for comparison, normed spaces are another generalization of finite-dimensional spaces that have many "nice" properties). It is precisely because $C_{c}^{\infty }\left(U\right)$  is a nuclear space that the Schwartz kernel theorem holds, as Alexander Grothendieck discovered when he investigated why the theorem works for the space of distributions but not for other "nice" spaces like the Hilbert space $L^{2}$  (this led him to discover nuclear maps and nuclear spaces, among other things). One of the primary results of the Schwartz kernel theorem is that for any open subsets $U_{1}\subseteq \mathbb {R} ^{m}$  and $U_{2}\subseteq \mathbb {R} ^{n}$ , the canonical map ${\mathcal {D}}^{\prime }\left(U_{1}\times U_{2}\right)\to L_{b}\left(C_{c}^{\infty }\left(U_{2}\right);{\mathcal {D}}^{\prime }\left(U_{1}\right)\right)$  is an isomorphism of TVSs (where $L_{b}\left(C_{c}^{\infty }\left(U_{2}\right);{\mathcal {D}}^{\prime }\left(U_{1}\right)\right)$  has the topology of uniform convergence on bounded subsets); this result is false if one replaces the space $C_{c}^{\infty }$  with $L^{2}$  (which is a reflexive space that is even isomorphic to its own strong dual space) and replaces ${\mathcal {D}}^{\prime }$  with the dual of this $L^{2}$  space.

Each of $C_{c}^{\infty }\left(U\right)$  and ${\mathcal {D}}^{\prime }\left(U\right)$  is a nuclear Montel space, which implies that they are each reflexive, barreled, Mackey, and have the Heine-Borel property. Other consequences are that:

• the topology on ${\mathcal {D}}^{\prime }\left(U\right)=\left(C_{c}^{\infty }\left(U\right)\right)_{b}^{\prime }$  (the strong dual topology) is identical to the topology of uniform convergence on compact subsets of $C_{c}^{\infty }\left(U\right)$ ;
• the strong dual space of ${\mathcal {D}}^{\prime }\left(U\right)$  is TVS isomorphic to $C_{c}^{\infty }\left(U\right)$  via the canonical TVS-isomorphism $C_{c}^{\infty }\left(U\right)\to \left({\mathcal {D}}^{\prime }\left(U\right)\right)_{b}^{\prime }$  defined by sending $f\in C_{c}^{\infty }\left(U\right)$  to value at $f$  (i.e. to the linear functional on ${\mathcal {D}}^{\prime }\left(U\right)$  defined by sending $d\in {\mathcal {D}}^{\prime }\left(U\right)$  to $d(f)$ );
• on any bounded subset of ${\mathcal {D}}^{\prime }\left(U\right)$ , the weak and strong subspace topologies coincide; the same is true for $C_{c}^{\infty }\left(U\right)$ ;
• every weakly convergent sequence in ${\mathcal {D}}^{\prime }\left(U\right)$  is strongly convergent (although this does not necessarily extend to nets).

#### Sequences of distributions

A sequence of distributions (Tk) converges with respect to the weak-* topology on D′(U) to a distribution T if and only if

$\langle T_{k},\varphi \rangle \to \langle T,\varphi \rangle$

for every test function $\varphi$  in D(U). For example, if fk : RR is the function

$f_{k}(x)={\begin{cases}k&{\text{if}}\ 0\leq x\leq 1/k\\0&{\text{otherwise}}\end{cases}}$

and Tk is the distribution corresponding to fk, then

$\langle T_{k},\varphi \rangle =k\int _{0}^{1/k}\varphi (x)\,dx\to \varphi (0)=\langle \delta ,\varphi \rangle$

as k → ∞, so Tkδ in D′(R). Thus, for large k, the function fk can be regarded as an approximation of the Dirac delta distribution.

## Localization of distributions

There is no way to define the value of a distribution in D′(U) at a particular point of U. However, as is the case with functions, distributions on U restrict to give distributions on open subsets of U. Furthermore, distributions are locally determined in the sense that a distribution on all of U can be assembled from a distribution on an open cover of U satisfying some compatibility conditions on the overlap. Such a structure is known as a sheaf.

### Restrictions to an open subset

Let U and V be open subsets of Rn with V ⊂ U. Let InVU : D(V) → D(U) be the operator which extends by zero a given smooth function compactly supported in V to a smooth function compactly supported in the larger set U. The transpose of InVU is called the restriction mapping and is denoted by $\rho _{VU}:={}^{t}\operatorname {In} _{VU}:{\mathcal {D}}^{\prime }(U)\to {\mathcal {D}}^{\prime }(V)$ .

The map InVU : D(V) → D(U) is a continuous injection where if $V\neq U$  then it is not a topological embedding and its range is not dense in D(U), which implies that this map's transpose is neither injective nor surjective and that the topology that InVU transfers from D(V) onto its image is strictly finer than the subspace topology that D(U) induces on this same set. A distribution S ∈ D′(V) is said to be extendible to U if it belongs to the range of the transpose of InVU and it is called extendible if it is extendable to $\mathbb {R} ^{n}$ .

For any distribution T ∈ D′(U), the restriction ρVU(T) is a distribution in the dual space D′(V) defined by

$\langle \rho _{VU}T,\varphi \rangle =\langle T,E_{VU}\varphi \rangle$

for all test functions $\varphi$  ∈ D(V).

Unless U = V, the restriction to V is neither injective nor surjective. Lack of surjectivity follows since distributions can blow up towards the boundary of V. For instance, if U = R and V = (0, 2), then the distribution

$T(x)=\sum _{n=1}^{\infty }n\,\delta \left(x-{\frac {1}{n}}\right)$

is in D′(V) but admits no extension to D′(U).

### Gluing and distributions that vanish in a set

Theorem Let I be a non-empty set and let $\left(U_{i}\right)_{i\in I}$  be a collection of open subsets of $\mathbb {R} ^{n}$ . For each $i\in I$ , let $T_{i}\in {\mathcal {D}}^{\prime }\left(U_{i}\right)$  and suppose that for all $i,j\in I$ , the restriction of $T_{i}$  to $U_{i}\cap U_{j}$  is equal to the restriction of $T_{j}$  to $U_{i}\cap U_{j}$  (note that both restrictions are elements of ${\mathcal {D}}^{\prime }\left(U_{i}\cap U_{j}\right)$ ). Then there exists a unique distribution $T\in {\mathcal {D}}^{\prime }\left(\cup _{i\in I}U_{i}\right)$  such that for all $i\in I$ , the restriction of T to $U_{i}$  is equal to $T_{i}$ .

If V is an open subset of U then we say that a distribution $T\in {\mathcal {D}}^{\prime }(U)$  vanishes in V if for each test function $f\in C_{c}^{\infty }\left(U\right)$ , if $\operatorname {supp} (f)\subseteq V$  then $Tf=0$ . T vanishes in V if and only if the restriction of T to V is equal to 0, or equivalently, if and only if T lies in the kernel of the restriction map ρVU.

Corollary: Let $\left(U_{i}\right)_{i\in I}$  be a collection of open subsets of $\mathbb {R} ^{n}$  and let $T\in {\mathcal {D}}^{\prime }\left(\cup _{i\in I}U_{i}\right)$ . Then T = 0 if and only if for each $i\in I$ , the restriction of T to $U_{i}$  is equal to 0.
Corollary: The union of all open subsets of U in which a distribution T vanishes is an open subset of U in which T vanishes.

### Support of a distribution

This last corollary implies that for every distribution T on U, there exists a unique largest subset V of U such that T vanishes in V (and doesn't vanish in any open subset of U that is not contained in V); the complement in U of this unique largest open subset is called the support of T. Thus

$\operatorname {supp} \,T=U\setminus \bigcup \left\{V\mid \rho _{VU}T=0\right\}.$

If f is a locally integrable function on U and if $D_{f}$  is its associated distribution, then the support of $D_{f}$  is the smallest closed subset of U in the complement of which f is almost everywhere equal to 0. If f is continuous, then the support of $D_{f}$  is equal to the closure of the set of points in U at which f doesn't vanish. The support of the distribution associated with the Dirac measure at a point $x_{0}$  is the set $\{x_{0}\}$ . If the support of a test function f does not intersect the support of a distribution T then Tf = 0. A distribution T is 0 if and only if its support is empty. If $f\in C^{\infty }\left(U\right)$  is identically 1 on some open set containing the support of a distribution T then the product f T = T. If the support of a distribution T is compact then it has finite order and furthermore, there is a constant C and a non-negative integer N such that

$\left|T\phi \right|\leq C\left\|\phi \right\|_{N}:=C\sup \left\{\left|\partial ^{\alpha }\phi (x)\right|:x\in U,|\alpha |\leq N\right\}$

for all $\phi \in C_{c}^{\infty }\left(U\right)$ . If T has compact support then it has a unique extension to a continuous linear functional ${\widehat {T}}$  on $C^{\infty }\left(U\right)$ ; this functional can be defined by ${\widehat {T}}f:=T\left(\psi f\right)$ , where $\psi \in C_{c}^{\infty }\left(U\right)$  is any function that is identically 1 on an open set containing the support of T.

If $S,T\in {\mathcal {D}}^{\prime }\left(U\right)$  and $\lambda \neq 0$  then $\operatorname {supp} \left(S+T\right)\subseteq \operatorname {supp} S\cup \operatorname {supp} T$  and $\operatorname {supp} \left(\lambda T\right)=\operatorname {supp} T$ . Thus, distributions with support in a given subset $A\subseteq U$  form a vector subspace of ${\mathcal {D}}^{\prime }\left(U\right)$ ; such a subspace is weakly closed in ${\mathcal {D}}^{\prime }\left(U\right)$  if and only if A is closed in U. Furthermore, if $P\left(x,\partial /\partial x\right)$  is a differential operator in U, then for all distributions T on U and all $f\in C^{\infty }\left(U\right)$  we have $\operatorname {supp} P\left(x,\partial /\partial x\right)T\subseteq \operatorname {supp} T$  and $\operatorname {supp} \left(fT\right)\subseteq \operatorname {supp} f\cap \operatorname {supp} T$ .

### Support in a point set and Dirac measures

For any $x\in U$ , let $\delta _{x}\in {\mathcal {D}}^{\prime }\left(U\right)$  denote the distribution induced by the Dirac measure at x. For any $x_{0}\in U$  and distribution $T\in {\mathcal {D}}^{\prime }\left(U\right)$ , the support of T is contained in $\{x_{0}\}$  if and only if T is a finite linear combination of derivatives of the Dirac measure at x, where if in addition the order of T is $\leq k$  (k a non-negative integer) then $T=\sum _{|p|\leq k}\alpha _{p}D^{p}\delta _{x}$  so some scalars $\alpha _{p}$  ($p$  ranges over all multi-indices such that $|p|\leq k$ ).

### Decomposition of distributions

Theorem Let $T\in {\mathcal {D}}^{\prime }\left(U\right)$ . There exists a sequence distributions $T_{1},T_{2},\ldots$  in ${\mathcal {D}}^{\prime }\left(U\right)$  such that each $T_{i}$  has compact support, every compact subset $K\subseteq U$  intersects the support of only finitely many $T_{i}$ 's, and the partial sums $\left(\sum _{i=1}^{n}T_{i}\right)_{n=1}^{\infty }$  converge in ${\mathcal {D}}^{\prime }\left(U\right)$  to T (i.e. $T=\sum _{i=1}^{\infty }T_{i}$ ).

## Operations on distributions

Many operations which are defined on smooth functions with compact support can also be defined for distributions. In general, if A : D(U) → D(U) is a linear mapping of vector spaces which is continuous with respect to the weak-* topology, then it is possible to extend A to a mapping A : D′(U) → D′(U) by passing to the limit. (This approach works for non-linear mappings as well, provided they are assumed to be uniformly continuous.)

In practice, however, it is more convenient to define operations on distributions by means of the transpose. If A : D(U) → D(U) is a continuous linear operator, then the transpose is an operator At : D(U) → D(U) such that for $\psi \in D(U)$ , $A^{t}\psi$  is the map $A^{t}\psi :U\to \mathbb {C}$  satisfying

$\int _{U}A\varphi (x)\cdot \psi (x)\,dx=\int _{U}\varphi (x)\cdot A^{t}\psi (x)\,dx\qquad {\text{for all}}\ \varphi \in D(U).$

(For operators acting on spaces of complex-valued test functions, the transpose At differs from the adjoint A* in that it does not include a complex conjugate.)

If such an operator At exists and is continuous on D(U), then the original operator A may be extended to D′(U) by defining AT for a distribution T as

$\langle AT,\varphi \rangle =\langle T,A^{t}\varphi \rangle \qquad {\text{for all}}\ \varphi \in D(U).$

### Differential operators

#### Differentiation

Suppose A : D(U) → D(U) is the partial derivative operator

$A\varphi ={\frac {\partial \varphi }{\partial x_{k}}}.$

If $\varphi$  and ψ are in D(U), then an integration by parts gives

$\int _{U}{\frac {\partial \varphi }{\partial x_{k}}}\psi \,dx=-\int _{U}\varphi {\frac {\partial \psi }{\partial x_{k}}}\,dx,$

so that At = −A. This operator is a continuous linear transformation on D(U). So, if T ∈ D′(U) is a distribution, then the partial derivative of T with respect to the coordinate xk is defined by the formula

$\left\langle {\frac {\partial T}{\partial x_{k}}},\varphi \right\rangle =-\left\langle T,{\frac {\partial \varphi }{\partial x_{k}}}\right\rangle \qquad {\text{for all}}\ \varphi \in D(U).$

With this definition, every distribution is infinitely differentiable, and the derivative in the direction xk is a linear operator on D′(U).

More generally, if α = (α1, ..., αn) is an arbitrary multi-index and ∂α is the associated partial derivative operator, then the partial derivative ∂αT of the distribution T ∈ D′(U) is defined by

$\left\langle \partial ^{\alpha }T,\varphi \right\rangle =(-1)^{|\alpha |}\left\langle T,\partial ^{\alpha }\varphi \right\rangle {\mbox{ for all }}\varphi \in \mathrm {D} (U).$

Differentiation of distributions is a continuous operator on D′(U); this is an important and desirable property that is not shared by most other notions of differentiation.

If T is a distribution in $\mathbb {R}$  then $\lim _{x\to 0}{\frac {T-\tau _{x}T}{x}}=DT$  in ${\mathcal {D}}^{\prime }\left(\mathbb {R} \right)$  where D T is the derivative of T and $\tau _{x}$  is translation by x; thus the derivative of T may be viewed as a limit of quotients.

#### Differential operators

A linear differential operator in U with C coefficients is a sum $P\left(x,\partial /\partial x\right):=\sum _{\alpha \in \mathbb {N} ^{n}}c_{\alpha }(x)\left(\partial /\partial x\right)^{\alpha }$  where all but finitely many of the C(U) functions $c_{\alpha }$  are identically 0. The integer $\sup \left\{\left|\alpha \right|:c_{\alpha }\neq 0\right\}$  is called the order of the differential operator $P\left(x,\partial /\partial x\right)$ , where $\left|\alpha \right|:=\alpha _{1}+\cdots +\alpha _{n}$  when $\alpha =\left(\alpha _{1},\ldots ,\alpha _{n}\right)$ . If $P\left(x,\partial /\partial x\right)$  is a linear differential operator of order k then it deduces a canonical linear map $C^{k}\left(U\right)\to C^{0}\left(U\right)$  defined by $\phi \mapsto P\left(x,\partial /\partial x\right)\phi$ , where we shall reuse notation and again denote this map by $P\left(x,\partial /\partial x\right)$ . The restriction of the canonical map to $C_{c}^{\infty }\left(U\right)$  induces a continuous linear map $P\left(x,\partial /\partial x\right){\big \vert }_{C_{c}^{\infty }\left(U\right)}:C_{c}^{\infty }\left(U\right)\to C_{c}^{\infty }\left(U\right)$  as well as another continuous linear map $P\left(x,\partial /\partial x\right){\big \vert }_{C^{\infty }\left(U\right)}:C^{\infty }\left(U\right)\to C^{\infty }\left(U\right)$ ; the transposes of these two map are consequently continuous linear maps ${\mathcal {D}}^{\prime }\left(U\right)\to {\mathcal {D}}^{\prime }\left(U\right)$  and ${\mathcal {E}}^{\prime }\left(U\right)\to {\mathcal {E}}^{\prime }\left(U\right)$ , respectively, where ${\mathcal {E}}^{\prime }\left(U\right)$  denotes the continuous dual space of $C^{\infty }\left(U\right)$ . Recall that the transpose of a continuous linear map $u:X\to Y$  is the map ${}^{t}u:Y^{\prime }\to X^{\prime }$  defined by ${}^{t}u\left(y^{\prime }\right):=y^{\prime }\circ u$ , or equivalently, it is the unique map satisfying $\left\langle y^{\prime },u(x)\right\rangle =\left\langle {}^{t}u\left(y^{\prime }\right),x\right\rangle$  for all $x\in X$  and all $y^{\prime }\in Y^{\prime }$ .

#### Differential operators acting on distributions and formal transposes

For any two complex-valued functions f and g on U, define $\left\langle f,g\right\rangle :=\int _{U}f(x)g(x)\operatorname {d} x$  (if the integral exists) and if $T\in {\mathcal {D}}^{\prime }\left(U\right)$  and $\phi \in C_{c}^{\infty }\left(U\right)$  then define $\left\langle T,\phi \right\rangle :=T(\phi )$ . Given any $\psi \in C^{\infty }\left(U\right)$ , let $D_{\psi }:C_{c}^{\infty }\left(U\right)\to \mathbb {C}$  be the canonical distribution defined by $D_{\psi }\left(\phi \right):=\int _{U}\phi (x)\psi (x)\operatorname {d} x$ .

We want to extend the action of a differential linear operator $P\left(x,\partial /\partial x\right)=\sum _{\alpha \in \mathbb {N} ^{n}}c_{\alpha }(x)\left(\partial /\partial x\right)^{\alpha }$  to distributions, where this extension (however it may be defined) will be the map denoted by ${\mathcal {D}}_{P\left(x,\partial /\partial x\right)}:{\mathcal {D}}^{\prime }\left(U\right)\to {\mathcal {D}}^{\prime }\left(U\right)$ . One property that this extension should reasonably be required to have is that ${\mathcal {D}}_{P\left(x,\partial /\partial x\right)}D_{\psi }=D_{P\left(x,\partial /\partial x\right)\psi }$  for any $\psi \in C^{\infty }\left(U\right)$ , where $P\left(x,\partial /\partial x\right)\psi$  is the smooth function that results from applying the differential operator $P\left(x,\partial /\partial x\right)$  to $\psi$ . It is natural to consider the transpose map ${}^{t}\left(P\left(x,\partial /\partial x\right){\big \vert }_{C_{c}^{\infty }\left(U\right)}\right):{\mathcal {D}}^{\prime }\left(U\right)\to {\mathcal {D}}^{\prime }\left(U\right)$  that $P\left(x,\partial /\partial x\right)$  canonically induces but this is not the extension of $P\left(x,\partial /\partial x\right)$  to distributions that we want because it doesn't have the aforementioned property. But by investigating this map's action on $D_{\psi }$ , we will be led to the appropriate definitions.

Note that for all $\phi \in C_{c}^{\infty }\left(U\right)$

$\left\langle {}^{t}P\left(x,\partial /\partial x\right)D_{\psi },\phi \right\rangle =\left\langle \psi (x),P\left(x,\partial /\partial x\right)\phi (x)\right\rangle =\int _{U}\psi (x)P\left(x,\partial /\partial x\right)\phi (x)\operatorname {d} x=\sum _{\alpha \in \mathbb {N} ^{n}}\int _{U}c_{\alpha }(x)\left[\left(\partial /\partial x\right)^{\alpha }\phi (x)\right]\psi (x)\operatorname {d} x$

Since $\phi$  has compact support, so too does every function $x\mapsto c_{\alpha }(x)\left[\left(\partial /\partial x\right)^{\alpha }\phi (x)\right]\psi (x)$  so when we integrate by parts we get

$\int _{U}c_{\alpha }(x)\left[\left(\partial /\partial x\right)^{\alpha }\phi (x)\right]\psi (x)\operatorname {d} x=\int _{U}\left(-1\right)^{|\alpha |}\left[\left(\partial /\partial x\right)^{\alpha }\left(c_{\alpha }(x)\psi (x)\right)\right]\phi (x)\operatorname {d} x$

(for instance, if $U=\mathbb {R}$ , $P\left(x,\partial /\partial x\right)={\frac {\partial }{\partial x}}$ , and the support of $\phi$  is contained in the interval [a, b], then $\int _{\mathbb {R} }\left({\frac {\partial }{\partial x}}\phi (x)\right)\psi (x)\operatorname {d} x=\int _{a}^{b}\left({\frac {\partial }{\partial x}}\phi (x)\right)\psi (x)\operatorname {d} x=\left(\phi (x)\psi (x)\right){\big \vert }_{a}^{b}-\int _{a}^{b}\left({\frac {\partial }{\partial x}}\psi (x)\right)\phi (x)\operatorname {d} x$ , where $\left(\phi (x)\psi (x)\right){\big \vert }_{a}^{b}=\phi (b)\psi (b)-\phi (a)\psi (a)=(0)\psi (b)-(0)\psi (a)=0$ ).

We now define the formal transpose of $P\left(x,\partial /\partial x\right)$ , denoted by ${}^{T}P\left(x,\partial /\partial x\right)$ , to be the differential operator in U defined by ${}^{T}P\left(x,\partial /\partial x\right):=\sum _{\alpha \in \mathbb {N} ^{n}}\left(-1\right)^{|\alpha |}b_{\alpha }(x)\left(\partial /\partial x\right)^{\alpha }$  where

$b_{\alpha }(x)=\sum _{\beta \geq \alpha }\left(-1\right)^{\left|\beta \right|}{\binom {\beta }{\alpha }}\left(\partial /\partial x\right)^{\beta -\alpha }c_{\beta }(x)$

with ${\binom {\beta }{\alpha }}:={\binom {\beta _{1}}{\alpha _{1}}}\cdots {\binom {\beta _{n}}{\alpha _{n}}}$  and with $\beta \geq \alpha$  if and only if $\beta _{i}\geq \alpha _{i}$  for all i. This definition stems from the fact that by using the Leibniz rule, we obtain

$\sum _{\alpha \in \mathbb {N} ^{n}}\int _{U}\left(-1\right)^{|\alpha |}\left[\left(\partial /\partial x\right)^{\alpha }\left(c_{\alpha }(x)\psi (x)\right)\right]\phi (x)\operatorname {d} x=\int _{U}\sum _{\alpha \in \mathbb {N} ^{n}}\left(-1\right)^{|\alpha |}\left[b_{\alpha }(x)\left(\partial /\partial x\right)^{\alpha }\psi (x)\right]\phi (x)\operatorname {d} x=\left\langle D_{{}^{T}P\left(x,\partial /\partial x\right)\psi },\phi \right\rangle$ .

We've thus shown that $\left\langle {}^{t}P\left(x,\partial /\partial x\right)D_{\psi },\phi \right\rangle =\left\langle D_{{}^{T}P\left(x,\partial /\partial x\right)\psi },\phi \right\rangle$  for all $\phi \in C_{c}^{\infty }\left(U\right)$  from which it follows that ${}^{t}P\left(x,\partial /\partial x\right)D_{\psi }=D_{{}^{T}P\left(x,\partial /\partial x\right)\psi }$ . Note that the formal transpose of the formal transpose is the original differential operator, i.e. ${}^{TT}P\left(x,\partial /\partial x\right)=P\left(x,\partial /\partial x\right)$ .

The formal transpose of $P\left(x,\partial /\partial x\right)$  induces a continuous linear map $\phi \mapsto {}^{T}P\left(x,\partial /\partial x\right)\phi$  from $C_{c}^{\infty }\left(U\right)$  into $C_{c}^{\infty }\left(U\right)$ , whose transpose will be denoted by $P\left(x,\partial /\partial x\right)$  or ${\mathcal {D}}_{P\left(x,\partial /\partial x\right)}$  or simply ${\mathcal {D}}_{P}$  and called a differential operator, where this map is a linear map ${\mathcal {D}}_{P}:{\mathcal {D}}^{\prime }\left(U\right)\to {\mathcal {D}}^{\prime }\left(U\right)$ . Explicitly, for any $T\in {\mathcal {D}}\left(U\right)$  and $\phi \in C_{c}^{\infty }\left(U\right)$ , ${\mathcal {D}}_{P}\left(T\right)\left(\phi \right)=T\left({}^{T}P\left(x,\partial /\partial x\right)\phi \right)$ . For any $\psi \in C^{\infty }\left(U\right)$ , we have ${\mathcal {D}}_{P}D_{\psi }=D_{P\left(x,\partial /\partial x\right)\psi }$ , which justifies our definition.

If $\left(T_{i}\right)_{i=1}^{\infty }$  converges to T in ${\mathcal {D}}^{\prime }\left(U\right)$  then for every multi-index $\alpha$ , $\left(\partial ^{\alpha }T_{i}\right)_{i=1}^{\infty }$  converges to $\partial ^{\alpha }T$  in ${\mathcal {D}}^{\prime }\left(U\right)$ .

##### Multiplication of a distribution by smooth function

Observe that a differential operator of degree 0 is just multiplication by a C function. And conversely, if $f\in C^{\infty }\left(U\right)$  then $P\left(x,\partial /\partial x\right):=f(x)$  is a differential operator of degree 0, whose formal transpose is itself (i.e. ${}^{T}P\left(x,\partial /\partial x\right)=P\left(x,\partial /\partial x\right)$ ). The induced differential operator ${\mathcal {D}}_{P}:{\mathcal {D}}^{\prime }\left(U\right)\to {\mathcal {D}}^{\prime }\left(U\right)$  maps a distribution T to a distribution denoted by $fT:={\mathcal {D}}_{P}(T)$ . We have thus defined the multiplication of a distribution by a $C^{\infty }$  function. If $\left(T_{i}\right)_{i=1}^{\infty }$  converges to T in ${\mathcal {D}}^{\prime }\left(U\right)$  and if $\left(f_{i}\right)_{i=1}^{\infty }$  converges to f in $C^{\infty }\left(U\right)$  then $\left(f_{i}T_{i}\right)_{i=1}^{\infty }$  converges to f T in ${\mathcal {D}}^{\prime }\left(U\right)$ .

We now give an alternative presentation of multiplication by a smooth function.

Multiplication by a smooth function

If m : UR is an infinitely differentiable function and T is a distribution on U, then the product mT is defined by

$\langle mT,\varphi \rangle =\langle T,m\varphi \rangle \qquad {\text{for all}}\ \varphi \in D(U).$

This definition coincides with the transpose definition since if M : D(U) → D(U) is the operator of multiplication by the function m (i.e., M$\varphi$  = m $\varphi$ ), then

$\int _{U}M\varphi (x)\cdot \psi (x)\,dx=\int _{U}m(x)\varphi (x)\cdot \psi (x)\,dx=\int _{U}\varphi (x)\cdot m(x)\psi (x)\,dx=\int _{U}\varphi (x)\cdot M\psi (x)\,dx,$

so that Mt = M.

Under multiplication by smooth functions, D′(U) is a module over the ring C(U). With this definition of multiplication by a smooth function, the ordinary product rule of calculus remains valid. However, a number of unusual identities also arise. For example, if δ is the Dirac delta distribution on R, then  = m(0)δ, and if δ′ is the derivative of the delta distribution, then

$m\delta '=m(0)\delta '-m'\delta =m(0)\delta '-m'(0)\delta .\,$

These definitions of differentiation and multiplication also make it possible to define the operation of a linear differential operator with smooth coefficients on a distribution. A linear differential operator P takes a distribution T ∈ D′(U) to another distribution PT given by a sum of the form

$PT=\sum \nolimits _{|\alpha |\leq k}p_{\alpha }\partial ^{\alpha }T,$

where the coefficients pα are smooth functions on U. The action of the distribution PT on a test function $\varphi$  is given by

$\left\langle \sum \nolimits _{|\alpha |\leq k}p_{\alpha }\partial ^{\alpha }T,\varphi \right\rangle =\left\langle T,\sum \nolimits _{|\alpha |\leq k}(-1)^{|\alpha |}\partial ^{\alpha }(p_{\alpha }\varphi )\right\rangle .$

The minimum integer k for which such an expansion holds for every distribution T is called the order of P. The space D′(U) is a D-module with respect to the action of the ring of linear differential operators.

The bilinear multiplication map $C^{\infty }\left(\mathbb {R} ^{n}\right)\times {\mathcal {D}}^{\prime }\left(\mathbb {R} ^{n}\right)\to {\mathcal {D}}^{\prime }\left(\mathbb {R} ^{n}\right)$  given by $\left(f,T\right)\mapsto fT$  is not continuous; it is however, hypocontinuous.

### Composition with a smooth function

Let T be a distribution on an open set U ⊂ Rn. Let V be an open set in Rn, and F : V → U. Then provided F is a submersion, it is possible to define

$T\circ F\in \mathrm {D} '(V).$

This is the composition of the distribution T with F, and is also called the pullback of T along F, sometimes written

$F^{\sharp }:T\mapsto F^{\sharp }T=T\circ F.$

The pullback is often denoted F*, although this notation should not be confused with the use of '*' to denote the adjoint of a linear mapping.

The condition that F be a submersion is equivalent to the requirement that the Jacobian derivative dF(x) of F is a surjective linear map for every x ∈ V. A necessary (but not sufficient) condition for extending F# to distributions is that F be an open mapping. The inverse function theorem ensures that a submersion satisfies this condition.

If F is a submersion, then F# is defined on distributions by finding the transpose map. Uniqueness of this extension is guaranteed since F# is a continuous linear operator on D(U). Existence, however, requires using the change of variables formula, the inverse function theorem (locally) and a partition of unity argument.

In the special case when F is a diffeomorphism from an open subset V of Rn onto an open subset U of Rn change of variables under the integral gives

$\int _{V}\varphi \circ F(x)\psi (x)\,dx=\int _{U}\varphi (x)\psi \left(F^{-1}(x)\right)\left|\det dF^{-1}(x)\right|\,dx.$

In this particular case, then, F# is defined by the transpose formula:

$\left\langle F^{\sharp }T,\varphi \right\rangle =\left\langle T,\left|\det d(F^{-1})\right|\varphi \circ F^{-1}\right\rangle .$

### 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 f and g are functions on $\mathbb {R} ^{n}$  then we denote by fg the convolution of f and g, defined at $x\in \mathbb {R} ^{n}$  to be the integral

$\left(f\ast g\right)(x):=\int _{\mathbb {R} ^{n}}f\left(x-y\right)g(y)\operatorname {d} y=\int _{\mathbb {R} ^{n}}f(y)g\left(x-y\right)\operatorname {d} y$

provided that the integral exists. If $1\leq p,q,r\leq \infty$  are such that 1/r = (1/p) + (1/q) - 1 then for any functions $f\in L^{p}\left(\mathbb {R} ^{n}\right)$  and $g\in L^{q}\left(\mathbb {R} ^{n}\right)$  we have $f\ast g\in L^{r}\left(\mathbb {R} ^{n}\right)$  and $\left\|f\ast g\right\|_{L^{r}}\leq \left\|f\right\|_{L^{p}}\left\|g\right\|_{L^{q}}$ . If f and g are continuous functions on $\mathbb {R} ^{n}$ , at least one of which has compact support, then $\operatorname {supp} (f\ast g)\subseteq \operatorname {supp} f+\operatorname {supp} g$  and if $A\subseteq \mathbb {R} ^{n}$  then the value of fg in the set A do not depend on the values of f outside of the Minkowski sum $A-\operatorname {supp} g=\left\{a-s:a\in A,s\in \operatorname {supp} g\right\}$ .

Importantly, if $g\in L^{1}\left(\mathbb {R} ^{n}\right)$  has compact support then for any $0\leq k\leq \infty$ , the convolution map $f\mapsto f\ast g$  is continuous when considered as the map $C^{k}\left(\mathbb {R} ^{n}\right)\to C^{k}\left(\mathbb {R} ^{n}\right)$  or as the map $C_{c}^{k}\left(\mathbb {R} ^{n}\right)\to C_{c}^{k}\left(\mathbb {R} ^{n}\right)$ .

Translation and symmetry

Given $a\in \mathbb {R} ^{a}$ , the translation operator τa is sends a function $f:\mathbb {R} ^{a}\to \mathbb {C}$  to the function $\tau _{a}f:\mathbb {R} ^{n}\to \mathbb {C}$  defined by $\tau _{a}f(y):=f(y-a)$ . This can be extended by the transpose to distributions in the following way: given a distribution T, the translation of T by a is the distribution $\tau _{a}T:C_{c}^{\infty }\left(\mathbb {R} ^{n}\right)\to \mathbb {C}$  defined by $\tau _{a}T\left(\varphi \right):=\left\langle T,\tau _{-a}\varphi \right\rangle$ .

Given a function $f:\mathbb {R} ^{a}\to \mathbb {C}$ , define the function ${\tilde {f}}:\mathbb {R} ^{a}\to \mathbb {C}$  by ${\tilde {f}}(x):=f\left(-x\right)$ . Given a distribution T, let ${\tilde {T}}:C_{c}^{\infty }\left(\mathbb {R} ^{n}\right)\to \mathbb {C}$  be the distribution defined by ${\tilde {T}}\left(\varphi \right):=T\left({\tilde {\varphi }}\right)$ . The operator $T\mapsto {\tilde {T}}$  is called the symmetry with respect to the origin.

#### Convolution of a smooth function with a distribution

Let $f\in C^{\infty }\left(\mathbb {R} ^{n}\right)$  and $T\in {\mathcal {D}}^{\prime }\left(\mathbb {R} ^{n}\right)$  and assume that at least one of f and T has compact support. For any $x\in \mathbb {R} ^{n}$ , note that $\tau _{x}{\tilde {f}}$  is the function $\tau _{x}{\tilde {f}}:\mathbb {R} ^{n}\to \mathbb {C}$  given by $y\mapsto \tau _{x}{\tilde {f}}(y)={\tilde {f}}\left(y-x\right)=f\left(x-y\right)$ . The convolution of f and T, denoted by $f\ast T$  or by $T\ast f$ , is the smooth function $f\ast T:\mathbb {R} ^{n}\to \mathbb {C}$  defined by $\left(f\ast T\right)(x):=\left\langle T,\tau _{x}{\tilde {f}}\right\rangle$ . We have $\operatorname {supp} (f\ast T)\subseteq \operatorname {supp} f+\operatorname {supp} T$ . For all multi-indices p, it satisfies $\left(\partial /\partial x\right)^{p}\left\langle T,\tau _{x}{\tilde {f}}\right\rangle =\left\langle T,\left(\partial /\partial x\right)^{p}\tau _{x}{\tilde {f}}\right\rangle$  and $\left(\partial /\partial x\right)^{p}\left(T\ast f\right)=\left(\left(\partial /\partial x\right)^{p}T\right)\ast f=T\ast \left(\left(\partial /\partial x\right)^{p}f\right)$ .

If T is a distribution then the map $f\mapsto T\ast f$  is continuous as a map $C_{c}^{\infty }\left(\mathbb {R} ^{n}\right)\to C^{\infty }\left(\mathbb {R} ^{n}\right)$  where if in addition T has compact support then it is also continuous as the map $C^{\infty }\left(\mathbb {R} ^{n}\right)\to C^{\infty }\left(\mathbb {R} ^{n}\right)$  and continuous as the map $C_{c}^{\infty }\left(\mathbb {R} ^{n}\right)\to C_{c}^{\infty }\left(\mathbb {R} ^{n}\right)$ .

If $L:C_{c}^{\infty }\left(\mathbb {R} ^{n}\right)\to C^{\infty }\left(\mathbb {R} ^{n}\right)$  is a continuous linear map such that $LD^{\alpha }\phi =D^{\alpha }L\phi$  for all $\alpha$  and all $\phi \in C_{c}^{\infty }\left(\mathbb {R} ^{n}\right)$  then there exists a distribution $T\in {\mathcal {D}}^{\prime }\left(\mathbb {R} ^{n}\right)$  such that $L\phi =T\circ \phi$  for all $\phi \in C_{c}^{\infty }\left(\mathbb {R} ^{n}\right)$ .

Example

Let H be the Heavyside function on $\mathbb {R}$  and $\delta$  be the Dirac measure at 0. Then $\left(D\delta \right)\ast H=\delta$ , where $D\delta$  is the derivative of $\delta$ , and $1\ast \left(D\delta \right)=0$ . Moreover, for any $\varphi \in C_{c}^{\infty }\left(\mathbb {R} \right)$ , $\left(H\ast \varphi \right)(x)=\int _{-\infty }^{x}\varphi (t)\operatorname {d} t$ . Importantly, the associative law fails to hold:

$1=1\ast \delta =1\ast \left(\left(D\delta \right)\ast H\right)$  but $\left(1\ast \left(D\delta \right)\right)\ast H=0\ast H=0$ .

#### Convolution of a test function with a distribution

If $f\in {\mathcal {D}}\left(\mathbb {R} ^{n}\right)$  is a compactly supported smooth test function, then convolution with f defines a linear map $C_{f}:{\mathcal {D}}\left(\mathbb {R} ^{n}\right)\to {\mathcal {D}}\left(\mathbb {R} ^{n}\right)$  given by $C_{f}\left(g\right):=f\ast g$ , which is continuous with respect to the canonical LF space topology on ${\mathcal {D}}\left(\mathbb {R} ^{n}\right)$ .

Convolution of f with a distribution $T\in {\mathcal {D}}^{\prime }\left(\mathbb {R} ^{n}\right)$  can be defined by taking the transpose of Cf relative to the duality pairing of ${\mathcal {D}}\left(\mathbb {R} ^{n}\right)$  with the space ${\mathcal {D}}^{\prime }\left(\mathbb {R} ^{n}\right)$  of distributions (Trèves 1967, Chapter 27). If fg$\varphi \in {\mathcal {D}}\left(\mathbb {R} ^{n}\right)$ , then by Fubini's theorem

$\left\langle C_{f}g,\varphi \right\rangle =\int _{\mathbf {R} ^{n}}\varphi (x)\int _{\mathbf {R} ^{n}}f(x-y)g(y)\,dy\,dx=\left\langle g,C_{\widetilde {f}}\varphi \right\rangle$

where ${{\widetilde {f}}(x)=f(-x)}$ . Extending by continuity, the convolution of f with a distribution T is defined by

$\langle f\ast T,\varphi \rangle =\left\langle T,{\widetilde {f}}\ast \varphi \right\rangle$

for all test functions $\varphi \in {\mathcal {D}}\left(\mathbb {R} ^{n}\right)$ .

An alternative way to define the convolution of a function f and a distribution T is to use the translation operator τa. The convolution of the compactly supported function f and the distribution T is then the function defined for each $x\in \mathbb {R} ^{n}$  by

$(f\ast T)(x)=\left\langle T,\tau _{x}{\widetilde {f}}\right\rangle .$

It can be shown that the convolution of a smooth, compactly supported function and a distribution is a smooth function. If the distribution T has compact support then if f is a polynomial (resp. an exponential function, an analytic function, the restriction to $\mathbb {R} ^{n}$  of an entire analytic function on $\mathbb {C} ^{n}$ , the restriction to $\mathbb {R} ^{n}$  of an entire function of exponential type in $\mathbb {C} ^{n}$ ) then the same is true of $T\ast f$ . If the distribution T has compact support as well, then fT is a compactly supported function, and the Titchmarsh convolution theorem (Hörmander 1983, Theorem 4.3.3) implies that

$\operatorname {ch} (\operatorname {supp} (f\ast T))=\operatorname {ch} \operatorname {supp} f+\operatorname {ch} \operatorname {supp} T$

where ch denotes the convex hull and supp denotes the support.

#### Convolution of distributions

It is also possible to define the convolution of two distributions S and T on $\mathbb {R} ^{n}$ , provided one of them has compact support. Informally, in order to define ST where T has compact support, the idea is to extend the definition of the convolution ∗ to a linear operation on distributions so that the associativity formula

$S\ast (T\ast \varphi )=(S\ast T)\ast \varphi$

continues to hold for all test functions $\varphi$ .

It is also possible to provide a more explicit characterization of the convolution of distributions (Trèves 1967, Chapter 27). Suppose that S and T are distributions and that S has compact support. Then the linear map $\bullet \ast {\tilde {S}}:C_{c}^{\infty }\left(\mathbb {R} ^{n}\right)\to C_{c}^{\infty }\left(\mathbb {R} ^{n}\right)$  defined by $f\mapsto f\ast {\tilde {S}}$  is continuous as is the linear map $\bullet \ast {\tilde {T}}:C_{c}^{\infty }\left(\mathbb {R} ^{n}\right)\to C^{\infty }\left(\mathbb {R} ^{n}\right)$  defined by $f\mapsto f\ast {\tilde {T}}$ . The transposes of these maps, ${}^{t}\left(\bullet \ast {\tilde {S}}\right):{\mathcal {D}}^{\prime }\left(\mathbb {R} ^{n}\right)\to {\mathcal {D}}^{\prime }\left(\mathbb {R} ^{n}\right)$  and ${}^{t}\left(\bullet \ast {\tilde {T}}\right):{\mathcal {E}}^{\prime }\left(\mathbb {R} ^{n}\right)\to {\mathcal {D}}^{\prime }\left(\mathbb {R} ^{n}\right)$ , are consequently continuous and one may show that ${}^{t}\left(\bullet \ast {\tilde {S}}\right)\left(T\right)={}^{t}\left(\bullet \ast {\tilde {T}}\right)\left(S\right)$ . This common value is called the convolution of S and T and it is a distribution that is denoted by $S\ast T$  or $T\ast S$ . It satisfies $\operatorname {supp} \left(S\ast T\right)\subseteq \operatorname {supp} S+\operatorname {supp} T$ . If S and T are two distributions, at least one of which has compact support, then for any $a\in \mathbb {R} ^{n}$ , $\tau _{a}\left(S\ast T\right)=\left(\tau _{a}S\right)\ast T=S\ast \left(\tau _{a}T\right)$