Distribution (mathematics)#Tempered distributions and Fourier transform

The practical use of distributions can be traced back to the use of Green's functions in the 1830s to solve ordinary differential equations, but was not formalized until much later. According to {{harvtxt|Kolmogorov|Fomin|1957}}, generalized functions originated in the work of {{harvs|txt|author-link=Sergei Lvovich Sobolev|first=Sergei|last= Sobolev|year=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. {{harvtxt|Gårding|1997}} comments that although the ideas in the transformative book by {{harvtxt|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. A detailed history of the theory of distributions was given by {{harvtxt|Lützen|1982}}.

The following notation will be used throughout this article:

• $n$ is a fixed positive integer and $U$ is a fixed non-empty open subset of Euclidean space $\backslash R^n.$
• $\backslash N\_0\; =\; \backslash \{0,\; 1,\; 2,\; \backslash ldots\backslash \}$ denotes the natural numbers.
• $k$ will denote a non-negative integer or $\backslash infty.$
• If $f$ is a function then $\backslash operatorname\{Dom\}(f)$ will denote its domain and the {{em|Support (mathematics)}} of $f,$ denoted by $\backslash operatorname\{supp\}(f),$ is defined to be the closure of the set $\backslash \{x\; \backslash in\; \backslash operatorname\{Dom\}(f):\; f(x)\; \backslash neq\; 0\backslash \}$ in $\backslash operatorname\{Dom\}(f).$
• For two functions $f,\; g\; :\; U\; \backslash to\; \backslash Complex,$ the following notation defines a canonical pairing: $$\backslash langle\; f,\; g\backslash rangle\; :=\; \backslash int\_U\; f(x)\; g(x)\; \backslash ,dx.$$
• A {{em|multi-index}} of size $n$ is an element in $\backslash N^n$ (given that $n$ is fixed, if the size of multi-indices is omitted then the size should be assumed to be $n$). The {{em|length}} of a multi-index $\backslash alpha\; =\; (\backslash alpha\_1,\; \backslash ldots,\; \backslash alpha\_n)\; \backslash in\; \backslash N^n$ is defined as $\backslash alpha\_1+\backslash cdots+\backslash alpha\_n$ and denoted by $|\backslash alpha|.$ Multi-indices are particularly useful when dealing with functions of several variables, in particular, we introduce the following notations for a given multi-index $\backslash alpha\; =\; (\backslash alpha\_1,\; \backslash ldots,\; \backslash alpha\_n)\; \backslash in\; \backslash N^n$: $$\backslash begin\{align\}$$

x^\alpha &= x_1^{\alpha_1} \cdots x_n^{\alpha_n} \\

\partial^\alpha &= \frac{\partial^

\end{align} We also introduce a partial order of all multi-indices by $\backslash beta\; \backslash ge\; \backslash alpha$ if and only if $\backslash beta\_i\; \backslash ge\; \backslash alpha\_i$ for all $1\; \backslash le\; i\backslash le\; n.$ When $\backslash beta\; \backslash ge\; \backslash alpha$ we define their multi-index binomial coefficient as: $$\backslash binom\{\backslash beta\}\{\backslash alpha\}\; :=\; \backslash binom\{\backslash beta\_1\}\{\backslash alpha\_1\}\; \backslash cdots\; \backslash binom\{\backslash beta\_n\}\{\backslash alpha\_n\}.$$

In this section, some basic notions and definitions needed to define real-valued distributions on {{mvar|U}} are introduced. Further discussion of the topologies on the spaces of test functions and distributions is given in the article on spaces of test functions and distributions.

{{block indent|em=1.5|text=Notation:

1. Let $k\; \backslash in\; \backslash \{0,\; 1,\; 2,\; \backslash ldots,\; \backslash infty\backslash \}.$
2. Let $C^k(U)$ denote the vector space of all {{mvar|k}}-times continuously differentiable real or complex-valued functions on {{mvar|U}}.
3. For any compact subset $K\; \backslash subseteq\; U,$ let $C^k(K)$ and $C^k(K;U)$ both denote the vector space of all those functions $f\; \backslash in\; C^k(U)$ such that $\backslash operatorname\{supp\}(f)\; \backslash subseteq\; K.$
4. * If $f\; \backslash in\; C^k(K)$ then the domain of $f$ is {{mvar|U}} and not {{mvar|K}}. So although $C^k(K)$ depends on both {{mvar|K}} and {{mvar|U}}, only {{mvar|K}} is typically indicated. The justification for this common practice is detailed below. The notation $C^k(K;U)$ will only be used when the notation $C^k(K)$ risks being ambiguous.
5. * Every $C^k(K)$ contains the constant {{math|0}} map, even if $K\; =\; \backslash varnothing.$
6. Let $C\_c^k(U)$ denote the set of all $f\; \backslash in\; C^k(U)$ such that $f\; \backslash in\; C^k(K)$ for some compact subset {{mvar|K}} of {{mvar|U}}.
7. * Equivalently, $C\_c^k(U)$ is the set of all $f\; \backslash in\; C^k(U)$ such that $f$ has compact support.
8. * $C\_c^k(U)$ is equal to the union of all $C^k(K)$ as $K\; \backslash subseteq\; U$ ranges over all compact subsets of $U.$
9. * If $f$ is a real-valued function on $U$, then $f$ is an element of $C\_c^k(U)$ if and only if $f$ is a $C^k$ bump function. Every real-valued test function on $U$ is also a complex-valued test function on $U.$

}}

File:Bump.png $(x,y)\; \backslash in\; \backslash R^2\; \backslash mapsto\; \backslash Psi(r),$ where $r\; =\; \backslash left(x^2\; +\; y^2\backslash right)^\backslash frac\{1\}\{2\}$ and This function is a test function on $\backslash R^2$ and is an element of $C^\backslash infty\_c\backslash left(\backslash R^2\backslash right).$ The support of this function is the closed unit disk in $\backslash R^2.$ It is non-zero on the open unit disk and it is equal to {{math|0}} everywhere outside of it.]]

For all $j,\; k\; \backslash in\; \backslash \{0,\; 1,\; 2,\; \backslash ldots,\; \backslash infty\backslash \}$ and any compact subsets $K$ and $L$ of $U$, we have:

$$\backslash begin\{align\}$$

C^k(K) &\subseteq C^k_c(U) \subseteq C^k(U) \\

C^k(K) &\subseteq C^k(L) && \text{if } K \subseteq L \\

C^k(K) &\subseteq C^j(K) && \text{if } j \le k \\

C_c^k(U) &\subseteq C^j_c(U) && \text{if } j \le k \\

C^k(U) &\subseteq C^j(U) && \text{if } j \le k \\

\end{align}

{{block indent|em=1.5|text=Definition: Elements of $C\_c^\backslash infty(U)$ are called {{em|test functions}} on {{mvar|U}} and $C\_c^\backslash infty(U)$ is called the {{em|space of test functions}} on {{mvar|U}}. We will use both $\backslash mathcal\{D\}(U)$ and $C\_c^\backslash infty(U)$ to denote this space.}}

Distributions on {{mvar|U}} are continuous linear functionals on $C\_c^\backslash infty(U)$ when this vector space is endowed with a particular topology called the {{em|canonical LF-topology}}. The following proposition states two necessary and sufficient conditions for the continuity of a linear function on $C\_c^\backslash infty(U)$ that are often straightforward to verify.

Proposition: A linear functional {{mvar|T}} on $C\_c^\backslash infty(U)$ is continuous, and therefore a {{em|distribution}}, if and only if any of the following equivalent conditions is satisfied:

1. For every compact subset $K\backslash subseteq\; U$ there exist constants $C>0$ and $N\backslash in\; \backslash N$ (dependent on $K$) such that for all $f\; \backslash in\; C\_c^\backslash infty(U)$ with support contained in $K$,{{sfn|Trèves|2006|pp=222-223}}{{harvnb|Grubb|2009|page=14}} $$|T(f)|\; \backslash leq\; C\; \backslash sup\; \backslash \{|\backslash partial^\backslash alpha\; f(x)|:\; x\; \backslash in\; U,\; |\backslash alpha|\; \backslash leq\; N\backslash \}.$$
2. For every compact subset $K\backslash subseteq\; U$ and every sequence $\backslash \{f\_i\backslash \}\_\{i=1\}^\backslash infty$ in $C\_c^\backslash infty(U)$ whose supports are contained in $K$, if $\backslash \{\backslash partial^\backslash alpha\; f\_i\backslash \}\_\{i=1\}^\backslash infty$ converges uniformly to zero on $U$ for every multi-index $\backslash alpha$, then $T(f\_i)\; \backslash to\; 0.$

We now introduce the seminorms that will define the topology on $C^k(U).$ Different authors sometimes use different families of seminorms so we list the most common families below. However, the resulting topology is the same no matter which family is used.

{{block indent|em=1.5|text=Suppose $k\; \backslash in\; \backslash \{0,\; 1,\; 2,\; \backslash ldots,\; \backslash infty\backslash \}$ and $K$ is an arbitrary compact subset of $U.$ Suppose $i$ is an integer such that $0\; \backslash leq\; i\; \backslash leq\; k$Note that $i$ being an integer implies $i\; \backslash neq\; \backslash infty.$ This is sometimes expressed as Since $\backslash infty\; +\; 1\; =\; \backslash infty,$ the inequality "" means: if $k\; =\; \backslash infty,$ while if $k\; \backslash neq\; \backslash infty$ then it means $0\; \backslash leq\; i\; \backslash leq\; k.$ and $p$ is a multi-index with length $|\; p|\backslash leq\; k.$ For $K\; \backslash neq\; \backslash varnothing$ and $f\; \backslash in\; C^k(U),$ define:

$$\backslash begin\{alignat\}\{4\}$$

\text{ (1) }\ & s_{p,K}(f) &&:= \sup_{x_0 \in K} \left| \partial^p f(x_0) \right| \\[4pt]

\text{ (2) }\ & q_{i,K}(f) &&:= \sup_{|p| \leq i} \left(\sup_{x_0 \in K} \left| \partial^p f(x_0) \right|\right) = \sup_{|p| \leq i} \left(s_{p, K}(f)\right) \\[4pt]

\text{ (3) }\ & r_{i,K}(f) &&:= \sup_{\stackrel{|p| \leq i}{x_0 \in K}} \left| \partial^p f(x_0) \right| \\[4pt]

\text{ (4) }\ & t_{i,K}(f) &&:= \sup_{x_0 \in K} \left(\sum_{|p| \leq i} \left| \partial^p f(x_0) \right|\right)

\end{alignat}

while for $K\; =\; \backslash varnothing,$ define all the functions above to be the constant {{math|0}} map.

}}

All of the functions above are non-negative $\backslash R$-valuedThe image of the compact set $K$ under a continuous $\backslash R$-valued map (for example, $x\; \backslash mapsto\; \backslash left|\backslash partial^p\; f(x)\backslash right|$ for $x\; \backslash in\; U$) is itself a compact, and thus bounded, subset of $\backslash R.$ If $K\; \backslash neq\; \backslash varnothing$ then this implies that each of the functions defined above is $\backslash R$-valued (that is, none of the supremums above are ever equal to $\backslash infty$). seminorms on $C^k(U).$ As explained in this article, every set of seminorms on a vector space induces a locally convex vector topology.

Each of the following sets of seminorms

$$\backslash begin\{alignat\}\{4\}$$

A ~:= \quad &\{q_{i,K} &&: \;K \text{ compact and } \;&&i \in \N \text{ satisfies } \;&&0 \leq i \leq k\} \\

B ~:= \quad &\{r_{i,K} &&: \;K \text{ compact and } \;&&i \in \N \text{ satisfies } \;&&0 \leq i \leq k\} \\

C ~:= \quad &\{t_{i,K} &&: \;K \text{ compact and } \;&&i \in \N \text{ satisfies } \;&&0 \leq i \leq k\} \\

D ~:= \quad &\{s_{p,K} &&: \;K \text{ compact and } \;&&p \in \N^n \text{ satisfies } \;&&|p| \leq k\}

\end{alignat}

generate the same locally convex vector topology on $C^k(U)$ (so for example, the topology generated by the seminorms in $A$ is equal to the topology generated by those in $C$).

{{block indent|em=1.5|text=The vector space $C^k(U)$ is endowed with the locally convex topology induced by any one of the four families $A,\; B,\; C,\; D$ of seminorms described above. This topology is also equal to the vector topology induced by {{em|all}} of the seminorms in $A\; \backslash cup\; B\; \backslash cup\; C\; \backslash cup\; D.$}}

With this topology, $C^k(U)$ becomes a locally convex Fréchet space that is {{em|not}} normable. Every element of $A\; \backslash cup\; B\; \backslash cup\; C\; \backslash cup\; D$ is a continuous seminorm on $C^k(U).$

Under this topology, a net $(f\_i)\_\{i\backslash in\; I\}$ in $C^k(U)$ converges to $f\; \backslash in\; C^k(U)$ if and only if for every multi-index $p$ with and every compact $K,$ the net of partial derivatives $\backslash left(\backslash partial^p\; f\_i\backslash right)\_\{i\; \backslash in\; I\}$ converges uniformly to $\backslash partial^p\; f$ on $K.${{sfn|Trèves|2006|pp=85-89}} For any $k\; \backslash in\; \backslash \{0,\; 1,\; 2,\; \backslash ldots,\; \backslash infty\backslash \},$ any (von Neumann) bounded subset of $C^\{k+1\}(U)$ is a relatively compact subset of $C^k(U).${{sfn|Trèves|2006|pp=142-149}} In particular, a subset of $C^\backslash infty(U)$ is bounded if and only if it is bounded in $C^i(U)$ for all $i\; \backslash in\; \backslash N.${{sfn|Trèves|2006| pp=142-149}} The space $C^k(U)$ is a Montel space if and only if $k\; =\; \backslash infty.${{sfn|Trèves|2006|pp=356-358}}

A subset $W$ of $C^\backslash infty(U)$ is open in this topology if and only if there exists $i\backslash in\; \backslash N$ such that $W$ is open when $C^\backslash infty(U)$ is endowed with the subspace topology induced on it by $C^i(U).$

As before, fix $k\; \backslash in\; \backslash \{0,\; 1,\; 2,\; \backslash ldots,\; \backslash infty\backslash \}.$ Recall that if $K$ is any compact subset of $U$ then $C^k(K)\; \backslash subseteq\; C^k(U).$

{{block indent|em=1.5|text=Assumption: For any compact subset $K\; \backslash subseteq\; U,$ we will henceforth assume that $C^k(K)$ is endowed with the subspace topology it inherits from the Fréchet space $C^k(U).$}}

If $k$ is finite then $C^k(K)$ is a Banach space{{sfn|Trèves|2006|pp=131-134}} with a topology that can be defined by the norm

{{anchor|Omitting the open set from notation}}

Suppose $U$ is an open subset of $\backslash R^n$ and $K\; \backslash subseteq\; U$ is a compact subset. By definition, elements of $C^k(K)$ are functions with domain $U$ (in symbols, $C^k(K)\; \backslash subseteq\; C^k(U)$), so the space $C^k(K)$ and its topology depend on $U;$ to make this dependence on the open set $U$ clear, temporarily denote $C^k(K)$ by $C^k(K;U).$

Importantly, changing the set $U$ to a different open subset $U\text{'}$ (with $K\; \backslash subseteq\; U\text{'}$) will change the set $C^k(K)$ from $C^k(K;U)$ to $C^k(K;U\text{'}),$Exactly as with $C^k(K;U),$ the space $C^k(K;\; U\text{'})$ is defined to be the vector subspace of $C^k(U\text{'})$ consisting of maps with support contained in $K$ endowed with the subspace topology it inherits from $C^k(U\text{'})$. so that elements of $C^k(K)$ will be functions with domain $U\text{'}$ instead of $U.$

Despite $C^k(K)$ depending on the open set ($U\; \backslash text\{\; or\; \}\; U\text{'}$), the standard notation for $C^k(K)$ makes no mention of it.

This is justified because, as this subsection will now explain, the space $C^k(K;U)$ is canonically identified as a subspace of $C^k(K;U\text{'})$ (both algebraically and topologically).

It is enough to explain how to canonically identify $C^k(K;\; U)$ with $C^k(K;\; U\text{'})$ when one of $U$ and $U\text{'}$ is a subset of the other. The reason is that if $V$ and $W$ are arbitrary open subsets of $\backslash R^n$ containing $K$ then the open set $U\; :=\; V\; \backslash cap\; W$ also contains $K,$ so that each of $C^k(K;\; V)$ and $C^k(K;\; W)$ is canonically identified with $C^k(K;\; V\; \backslash cap\; W)$ and now by transitivity, $C^k(K;\; V)$ is thus identified with $C^k(K;\; W).$

So assume $U\; \backslash subseteq\; V$ are open subsets of $\backslash R^n$ containing $K.$

Given $f\; \backslash in\; C\_c^k(U),$ its {{em|trivial extension to $V$}} is the function $F\; :\; V\; \backslash to\; \backslash Complex$ defined by:

$$F(x)\; =\; \backslash begin\{cases\}$$

f(x) & x \in U, \\

0 & \text{otherwise}.

\end{cases}

This trivial extension belongs to $C^k(V)$ (because $f\; \backslash in\; C\_c^k(U)$ has compact support) and it will be denoted by $I(f)$ (that is, $I(f)\; :=\; F$). The assignment $f\; \backslash mapsto\; I(f)$ thus induces a map $I\; :\; C\_c^k(U)\; \backslash to\; C^k(V)$ that sends a function in $C\_c^k(U)$ to its trivial extension on $V.$ This map is a linear injection and for every compact subset $K\; \backslash subseteq\; U$ (where $K$ is also a compact subset of $V$ since $K\; \backslash subseteq\; U\; \backslash subseteq\; V$),

$$\backslash begin\{alignat\}\{4\}$$

I\left(C^k(K; U)\right) &~=~ C^k(K; V) \qquad \text{ and thus } \\

I\left(C_c^k(U)\right) &~\subseteq~ C_c^k(V).

\end{alignat}

If $I$ is restricted to $C^k(K;\; U)$ then the following induced linear map is a homeomorphism (linear homeomorphisms are called {{em|TVS-isomorphisms}}):

$$\backslash begin\{alignat\}\{4\}$$

\,& C^k(K; U) && \to \,&& C^k(K;V) \\

& f && \mapsto\,&& I(f) \\

\end{alignat}

and thus the next map is a topological embedding:

$$\backslash begin\{alignat\}\{4\}$$

\,& C^k(K; U) && \to \,&& C^k(V) \\

& f && \mapsto\,&& I(f). \\

\end{alignat}

Using the injection

$$I\; :\; C\_c^k(U)\; \backslash to\; C^k(V)$$

the vector space $C\_c^k(U)$ is canonically identified with its image in $C\_c^k(V)\; \backslash subseteq\; C^k(V).$ Because $C^k(K;\; U)\; \backslash subseteq\; C\_c^k(U),$ through this identification, $C^k(K;\; U)$ can also be considered as a subset of $C^k(V).$

Thus the topology on $C^k(K;U)$ is independent of the open subset $U$ of $\backslash R^n$ that contains $K,${{sfn|Rudin|1991|pp=149-181}} which justifies the practice of writing $C^k(K)$ instead of $C^k(K;\; U).$

{{Main|Spaces of test functions and distributions}}

{{See also|LF-space|Topology of uniform convergence}}

Recall that $C\_c^k(U)$ denotes all functions in $C^k(U)$ that have compact support in $U,$ where note that $C\_c^k(U)$ is the union of all $C^k(K)$ as $K$ ranges over all compact subsets of $U.$ Moreover, for each $k,\backslash ,\; C\_c^k(U)$ is a dense subset of $C^k(U).$ The special case when $k\; =\; \backslash infty$ gives us the space of test functions.

{{block indent|em=1.5|text=$C\_c^\backslash infty(U)$ is called the {{em|space of test functions on $U$}} and it may also be denoted by $\backslash mathcal\{D\}(U).$ Unless indicated otherwise, it is endowed with a topology called {{em|the canonical LF topology}}, whose definition is given in the article: Spaces of test functions and distributions.}}

The canonical LF-topology is {{em|not}} metrizable and importantly, it is Comparison of topologies than the subspace topology that $C^\backslash infty(U)$ induces on $C\_c^\backslash infty(U).$ However, the canonical LF-topology does make $C\_c^\backslash infty(U)$ into a complete reflexive nuclear{{sfn|Trèves|2006|pp=526-534}} Montel{{sfn|Trèves|2006|p=357}} bornological barrelled Mackey space; the same is true of its strong dual space (that is, the space of all distributions with its usual topology). The canonical LF-topology can be defined in various ways.

{{See also|Continuous linear functional}}

As discussed earlier, continuous linear functionals on a $C\_c^\backslash infty(U)$ are known as distributions on $U.$ Other equivalent definitions are described below.

{{block indent|em=1.5|text=By definition, a {{em|distribution on $U$}} is a continuous linear functional on $C\_c^\backslash infty(U).$ Said differently, a distribution on $U$ is an element of the continuous dual space of $C\_c^\backslash infty(U)$ when $C\_c^\backslash infty(U)$ is endowed with its canonical LF topology.}}

There is a canonical duality pairing between a distribution $T$ on $U$ and a test function $f\; \backslash in\; C\_c^\backslash infty(U),$ which is denoted using angle brackets by

$$\backslash begin\{cases\}$$

\mathcal{D}'(U) \times C_c^\infty(U) \to \R \\

(T, f) \mapsto \langle T, f \rangle := T(f)

\end{cases}

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 $T.$

Proposition. If $T$ is a linear functional on $C\_c^\backslash infty(U)$ then the following are equivalent:

1. {{mvar|T}} is a distribution;
2. {{mvar|T}} is continuous;
3. {{mvar|T}} is continuous at the origin;
4. {{mvar|T}} is uniformly continuous;
5. {{mvar|T}} is a bounded operator;
6. {{mvar|T}} is sequentially continuous;
7. * explicitly, for every sequence $\backslash left(f\_i\backslash right)\_\{i=1\}^\backslash infty$ in $C\_c^\backslash infty(U)$ that converges in $C\_c^\backslash infty(U)$ to some $f\; \backslash in\; C\_c^\backslash infty(U),$ $\backslash lim\_\{i\; \backslash to\; \backslash infty\}\; T\backslash left(f\_i\backslash right)\; =\; T(f);$Even though the topology of $C\_c^\backslash infty(U)$ is not metrizable, a linear functional on $C\_c^\backslash infty(U)$ is continuous if and only if it is sequentially continuous.
8. {{mvar|T}} is sequentially continuous at the origin; in other words, {{mvar|T}} maps null sequencesA {{em|null sequence}} is a sequence that converges to the origin. to null sequences;
9. * explicitly, for every sequence $\backslash left(f\_i\backslash right)\_\{i=1\}^\backslash infty$ in $C\_c^\backslash infty(U)$ that converges in $C\_c^\backslash infty(U)$ to the origin (such a sequence is called a {{em|null sequence}}), $\backslash lim\_\{i\; \backslash to\; \backslash infty\}\; T\backslash left(f\_i\backslash right)\; =\; 0;$
10. * a {{em|null sequence}} is by definition any sequence that converges to the origin;
11. {{mvar|T}} maps null sequences to bounded subsets;
12. * explicitly, for every sequence $\backslash left(f\_i\backslash right)\_\{i=1\}^\backslash infty$ in $C\_c^\backslash infty(U)$ that converges in $C\_c^\backslash infty(U)$ to the origin, the sequence $\backslash left(T\backslash left(f\_i\backslash right)\backslash right)\_\{i=1\}^\backslash infty$ is bounded;
13. {{mvar|T}} maps Mackey convergent null sequences to bounded subsets;
14. * explicitly, for every Mackey convergent null sequence $\backslash left(f\_i\backslash right)\_\{i=1\}^\backslash infty$ in $C\_c^\backslash infty(U),$ the sequence $\backslash left(T\backslash left(f\_i\backslash right)\backslash right)\_\{i=1\}^\backslash infty$ is bounded;
15. * a sequence $f\_\{\backslash bull\}\; =\; \backslash left(f\_i\backslash right)\_\{i=1\}^\backslash infty$ is said to be {{em|Mackey convergent to the origin}} if there exists a divergent sequence $r\_\{\backslash bull\}\; =\; \backslash left(r\_i\backslash right)\_\{i=1\}^\backslash infty\; \backslash to\; \backslash infty$ of positive real numbers such that the sequence $\backslash left(r\_i\; f\_i\backslash right)\_\{i=1\}^\backslash infty$ is bounded; every sequence that is Mackey convergent to the origin necessarily converges to the origin (in the usual sense);
16. The kernel of {{mvar|T}} is a closed subspace of $C\_c^\backslash infty(U);$
17. The graph of {{mvar|T}} is closed;

1. There exists a continuous seminorm $g$ on $C\_c^\backslash infty(U)$ such that $|T|\; \backslash leq\; g;$
2. There exists a constant $C\; >\; 0$ and a finite subset $\backslash \{g\_1,\; \backslash ldots,\; g\_m\backslash \}\; \backslash subseteq\; \backslash mathcal\{P\}$ (where $\backslash mathcal\{P\}$ is any collection of continuous seminorms that defines the canonical LF topology on $C\_c^\backslash infty(U)$) such that $|T|\; \backslash leq\; C(g\_1\; +\; \backslash cdots\; +\; g\_m);$If $\backslash mathcal\{P\}$ is also directed under the usual function comparison then we can take the finite collection to consist of a single element.
3. For every compact subset $K\backslash subseteq\; U$ there exist constants $C>0$ and $N\backslash in\; \backslash N$ such that for all $f\; \backslash in\; C^\backslash infty(K),${{sfn|Trèves|2006|pp=222-223}} $$|T(f)|\; \backslash leq\; C\; \backslash sup\; \backslash \{|\backslash partial^\backslash alpha\; f(x)|\; :\; x\; \backslash in\; U,\; |\backslash alpha|\backslash leq\; N\backslash \};$$
4. For every compact subset $K\backslash subseteq\; U$ there exist constants $C\_K>0$ and $N\_K\backslash in\; \backslash N$ such that for all $f\; \backslash in\; C\_c^\backslash infty(U)$ with support contained in $K,$See for example {{harvnb|Grubb|2009|page=14}}. $$|T(f)|\; \backslash leq\; C\_K\; \backslash sup\; \backslash \{|\backslash partial^\backslash alpha\; f(x)|\; :\; x\; \backslash in\; K,\; |\backslash alpha|\backslash leq\; N\_K\backslash \};$$
5. For any compact subset $K\backslash subseteq\; U$ and any sequence $\backslash \{f\_i\backslash \}\_\{i=1\}^\backslash infty$ in $C^\backslash infty(K),$ if $\backslash \{\backslash partial^p\; f\_i\backslash \}\_\{i=1\}^\backslash infty$ converges uniformly to zero for all multi-indices $p,$ then $T(f\_i)\; \backslash to\; 0;$

The set of all distributions on $U$ is the continuous dual space of $C\_c^\backslash infty(U),$ which when endowed with the strong dual topology is denoted by $\backslash mathcal\{D\}\text{'}(U).$ Importantly, unless indicated otherwise, the topology on $\backslash mathcal\{D\}\text{'}(U)$ is the strong dual topology; if the topology is instead the weak-* topology then this will be indicated. Neither topology is metrizable although unlike the weak-* topology, the strong dual topology makes $\backslash mathcal\{D\}\text{'}(U)$ into a complete nuclear space, to name just a few of its desirable properties.

Neither $C\_c^\backslash infty(U)$ nor its strong dual $\backslash mathcal\{D\}\text{'}(U)$ is a sequential space and so neither of their topologies can be fully described by sequences (in other words, defining only what sequences converge in these spaces is {{em|not}} enough to fully/correctly define their topologies).

However, a {{em|sequence}} in $\backslash mathcal\{D\}\text{'}(U)$ converges in the strong dual topology if and only if it converges in the weak-* topology (this leads many authors to use pointwise convergence to {{em|define}} the convergence of a sequence of distributions; this is fine for sequences but this is {{em|not}} guaranteed to extend to the convergence of nets of distributions because a net may converge pointwise but fail to converge in the strong dual topology).

More information about the topology that $\backslash mathcal\{D\}\text{'}(U)$ is endowed with can be found in the article on spaces of test functions and distributions and the articles on polar topologies and dual systems.

A Linear map from $\backslash mathcal\{D\}\text{'}(U)$ into another locally convex topological vector space (such as any normed space) is continuous if and only if it is sequentially continuous at the origin. However, this is no longer guaranteed if the map is not linear or for maps valued in more general topological spaces (for example, that are not also locally convex topological vector spaces). The same is true of maps from $C\_c^\backslash infty(U)$ (more generally, this is true of maps from any locally convex bornological space).

There is no way to define the value of a distribution in $\backslash mathcal\{D\}\text{'}(U)$ at a particular point of {{mvar|U}}. However, as is the case with functions, distributions on {{mvar|U}} restrict to give distributions on open subsets of {{mvar|U}}. Furthermore, distributions are {{em|locally determined}} in the sense that a distribution on all of {{mvar|U}} can be assembled from a distribution on an open cover of {{mvar|U}} satisfying some compatibility conditions on the overlaps. Such a structure is known as a sheaf.

Let $V\; \backslash subseteq\; U$ be open subsets of $\backslash R^n.$

Every function $f\; \backslash in\; \backslash mathcal\{D\}(V)$ can be {{em|extended by zero}} from its domain {{mvar|V}} to a function on {{mvar|U}} by setting it equal to $0$ on the complement $U\; \backslash setminus\; V.$ This extension is a smooth compactly supported function called the {{em|trivial extension of $f$ to $U$}} and it will be denoted by $E\_\{VU\}\; (f).$

This assignment $f\; \backslash mapsto\; E\_\{VU\}\; (f)$ defines the {{em|trivial extension}} operator

$E\_\{VU\}\; :\; \backslash mathcal\{D\}(V)\; \backslash to\; \backslash mathcal\{D\}(U),$

which is a continuous injective linear map. It is used to canonically identify $\backslash mathcal\{D\}(V)$ as a vector subspace of $\backslash mathcal\{D\}(U)$ (although {{em|not}} as a topological subspace).

Its transpose (explained here)

$$\backslash rho\_\{VU\}\; :=\; \{\}^\{t\}E\_\{VU\}\; :\; \backslash mathcal\{D\}\text{'}(U)\; \backslash to\; \backslash mathcal\{D\}\text{'}(V),$$

is called the {{em|{{visible anchor|restriction map|text=restriction to $V$ of distributions in $U$}}}}{{sfn|Trèves|2006|pp=245-247}} and as the name suggests, the image $\backslash rho\_\{VU\}(T)$ of a distribution $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(U)$ under this map is a distribution on $V$ called the restriction of $T$ to $V.$ The defining condition of the restriction $\backslash rho\_\{VU\}(T)$ is:

$$\backslash langle\; \backslash rho\_\{VU\}\; T,\; \backslash phi\; \backslash rangle\; =\; \backslash langle\; T,\; E\_\{VU\}\; \backslash phi\; \backslash rangle\; \backslash quad\; \backslash text\{\; for\; all\; \}\; \backslash phi\; \backslash in\; \backslash mathcal\{D\}(V).$$

If $V\; \backslash neq\; U$ then the (continuous injective linear) trivial extension map $E\_\{VU\}\; :\; \backslash mathcal\{D\}(V)\; \backslash to\; \backslash mathcal\{D\}(U)$ is {{em|not}} a topological embedding (in other words, if this linear injection was used to identify $\backslash mathcal\{D\}(V)$ as a subset of $\backslash mathcal\{D\}(U)$ then $\backslash mathcal\{D\}(V)$'s topology would strictly finer than the subspace topology that $\backslash mathcal\{D\}(U)$ induces on it; importantly, it would {{em|not}} be a topological subspace since that requires equality of topologies) and its range is also {{em|not}} dense in its codomain $\backslash mathcal\{D\}(U).${{sfn|Trèves|2006|pp=245-247}} Consequently if $V\; \backslash neq\; U$ then the restriction mapping is neither injective nor surjective.{{sfn|Trèves|2006|pp=245-247}} A distribution $S\; \backslash in\; \backslash mathcal\{D\}\text{'}(V)$ is said to be {{em|extendible to {{mvar|U}}}} if it belongs to the range of the transpose of $E\_\{VU\}$ and it is called {{em|extendible}} if it is extendable to $\backslash R^n.${{sfn|Trèves|2006|pp=245-247}}

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

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

is in $\backslash mathcal\{D\}\text{'}(V)$ but admits no extension to $\backslash mathcal\{D\}\text{'}(U).$

{{Math theorem

| name = Theorem{{sfn |Trèves|2006|pp=253-255}}

| math_statement = Let $(U\_i)\_\{i\; \backslash in\; I\}$ be a collection of open subsets of $\backslash R^n.$ For each $i\; \backslash in\; I,$ let $T\_i\; \backslash in\; \backslash mathcal\{D\}\text{'}(U\_i)$ and suppose that for all $i,\; j\; \backslash in\; I,$ the restriction of $T\_i$ to $U\_i\; \backslash cap\; U\_j$ is equal to the restriction of $T\_j$ to $U\_i\; \backslash cap\; U\_j$ (note that both restrictions are elements of $\backslash mathcal\{D\}\text{'}(U\_i\; \backslash cap\; U\_j)$). Then there exists a unique $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(\backslash bigcup\_\{i\; \backslash in\; I\}\; U\_i)$ such that for all $i\; \backslash in\; I,$ the restriction of {{mvar|T}} to $U\_i$ is equal to $T\_i.$

}}

Let {{mvar|V}} be an open subset of {{mvar|U}}. $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(U)$ is said to {{em|vanish in {{mvar|V}}}} if for all $f\; \backslash in\; \backslash mathcal\{D\}(U)$ such that $\backslash operatorname\{supp\}(f)\; \backslash subseteq\; V$ we have $Tf\; =\; 0.$ {{mvar|T}} vanishes in {{mvar|V}} if and only if the restriction of {{mvar|T}} to {{mvar|V}} is equal to 0, or equivalently, if and only if {{mvar|T}} lies in the kernel of the restriction map $\backslash rho\_\{VU\}.$

{{Math theorem

| name = Corollary{{sfn |Trèves|2006| pp=253-255}}

| math_statement = Let $(U\_i)\_\{i\; \backslash in\; I\}$ be a collection of open subsets of $\backslash R^n$ and let $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(\backslash bigcup\_\{i\; \backslash in\; I\}\; U\_i).$ $T\; =\; 0$ if and only if for each $i\; \backslash in\; I,$ the restriction of {{mvar|T}} to $U\_i$ is equal to 0.

}}

{{Math theorem| name=Corollary{{sfn |Trèves|2006|pp=253-255}}| math_statement= The union of all open subsets of {{mvar|U}} in which a distribution {{mvar|T}} vanishes is an open subset of {{mvar|U}} in which {{mvar|T}} vanishes.}}

This last corollary implies that for every distribution {{mvar|T}} on {{mvar|U}}, there exists a unique largest subset {{mvar|V}} of {{mvar|U}} such that {{mvar|T}} vanishes in {{mvar|V}} (and does not vanish in any open subset of {{mvar|U}} that is not contained in {{mvar|V}}); the complement in {{mvar|U}} of this unique largest open subset is called {{em|the support of {{mvar|T}}}}.{{sfn|Trèves|2006|pp=253-255}} Thus

$$\backslash operatorname\{supp\}(T)\; =\; U\; \backslash setminus\; \backslash bigcup\; \backslash \{V\; \backslash mid\; \backslash rho\_\{VU\}T\; =\; 0\backslash \}.$$

If $f$ is a locally integrable function on {{mvar|U}} and if $D\_f$ is its associated distribution, then the support of $D\_f$ is the smallest closed subset of {{mvar|U}} in the complement of which $f$ is almost everywhere equal to 0.{{sfn|Trèves|2006|pp=253-255}} If $f$ is continuous, then the support of $D\_f$ is equal to the closure of the set of points in {{mvar|U}} at which $f$ does not vanish.{{sfn|Trèves|2006| pp=253-255}} The support of the distribution associated with the Dirac measure at a point $x\_0$ is the set $\backslash \{x\_0\backslash \}.${{sfn|Trèves|2006|pp=253-255}} If the support of a test function $f$ does not intersect the support of a distribution {{mvar|T}} then $Tf\; =\; 0.$ A distribution {{mvar|T}} is 0 if and only if its support is empty. If $f\; \backslash in\; C^\backslash infty(U)$ is identically 1 on some open set containing the support of a distribution {{mvar|T}} then $f\; T\; =\; T.$ If the support of a distribution {{mvar|T}} is compact then it has finite order and there is a constant $C$ and a non-negative integer $N$ such that:{{sfn|Rudin|1991|pp=149-181}}

$$|T\; \backslash phi|\; \backslash leq\; C\backslash |\backslash phi\backslash |\_N\; :=\; C\; \backslash sup\; \backslash left\backslash \{\backslash left|\backslash partial^\backslash alpha\; \backslash phi(x)\backslash right|\; :\; x\; \backslash in\; U,\; |\backslash alpha|\; \backslash leq\; N\; \backslash right\backslash \}\; \backslash quad\; \backslash text\{\; for\; all\; \}\; \backslash phi\; \backslash in\; \backslash mathcal\{D\}(U).$$

If {{mvar|T}} has compact support, then it has a unique extension to a continuous linear functional $\backslash widehat\{T\}$ on $C^\backslash infty(U)$; this function can be defined by $\backslash widehat\{T\}\; (f)\; :=\; T(\backslash psi\; f),$ where $\backslash psi\; \backslash in\; \backslash mathcal\{D\}(U)$ is any function that is identically 1 on an open set containing the support of {{mvar|T}}.{{sfn|Rudin|1991|pp=149-181}}

If $S,\; T\; \backslash in\; \backslash mathcal\{D\}\text{'}(U)$ and $\backslash lambda\; \backslash neq\; 0$ then $\backslash operatorname\{supp\}(S\; +\; T)\; \backslash subseteq\; \backslash operatorname\{supp\}(S)\; \backslash cup\; \backslash operatorname\{supp\}(T)$ and $\backslash operatorname\{supp\}(\backslash lambda\; T)\; =\; \backslash operatorname\{supp\}(T).$ Thus, distributions with support in a given subset $A\; \backslash subseteq\; U$ form a vector subspace of $\backslash mathcal\{D\}\text{'}(U).${{sfn|Trèves|2006|pp=255-257}} Furthermore, if $P$ is a differential operator in {{mvar|U}}, then for all distributions {{mvar|T}} on {{mvar|U}} and all $f\; \backslash in\; C^\backslash infty(U)$ we have $\backslash operatorname\{supp\}\; (P(x,\; \backslash partial)\; T)\; \backslash subseteq\; \backslash operatorname\{supp\}(T)$ and $\backslash operatorname\{supp\}(fT)\; \backslash subseteq\; \backslash operatorname\{supp\}(f)\; \backslash cap\; \backslash operatorname\{supp\}(T).${{sfn|Trèves|2006|pp=255-257}}

For any $x\; \backslash in\; U,$ let $\backslash delta\_x\; \backslash in\; \backslash mathcal\{D\}\text{'}(U)$ denote the distribution induced by the Dirac measure at $x.$ For any $x\_0\; \backslash in\; U$ and distribution $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(U),$ the support of {{mvar|T}} is contained in $\backslash \{x\_0\backslash \}$ if and only if {{mvar|T}} is a finite linear combination of derivatives of the Dirac measure at $x\_0.${{sfn|Trèves|2006|pp=264-266}} If in addition the order of {{mvar|T}} is $\backslash leq\; k$ then there exist constants $\backslash alpha\_p$ such that:{{sfn|Rudin|1991|p=165}}

$$T\; =\; \backslash sum\_\{|p|\; \backslash leq\; k\}\; \backslash alpha\_p\; \backslash partial^p\; \backslash delta\_\{x\_0\}.$$

Said differently, if {{mvar|T}} has support at a single point $\backslash \{P\backslash \},$ then {{mvar|T}} is in fact a finite linear combination of distributional derivatives of the $\backslash delta$ function at {{mvar|P}}. That is, there exists an integer {{mvar|m}} and complex constants $a\_\backslash alpha$ such that

$$T\; =\; \backslash sum\_\{|\backslash alpha|\backslash leq\; m\}\; a\_\backslash alpha\; \backslash partial^\backslash alpha(\backslash tau\_P\backslash delta)$$

where $\backslash tau\_P$ is the translation operator.

{{Math theorem|name=Theorem{{sfn|Rudin|1991|pp=149-181}}|math_statement=

Suppose {{mvar|T}} is a distribution on {{mvar|U}} with compact support {{mvar|K}}. There exists a continuous function $f$ defined on {{mvar|U}} and a multi-index {{math|1=p}} such that

$$T\; =\; \backslash partial^p\; f,$$

where the derivatives are understood in the sense of distributions. That is, for all test functions $\backslash phi$ on {{mvar|U}},

$$T\; \backslash phi\; =\; (-1)^$$

}}

{{Math theorem|name=Theorem{{sfn|Rudin|1991|pp=149-181}}|math_statement=

Suppose {{mvar|T}} is a distribution on {{mvar|U}} with compact support {{mvar|K}} and let {{mvar|V}} be an open subset of {{mvar|U}} containing {{mvar|K}}. Since every distribution with compact support has finite order, take {{mvar|N}} to be the order of {{mvar|T}} and define $P:=\backslash \{0,1,\backslash ldots,\; N+2\backslash \}^n.$ There exists a family of continuous functions $(f\_p)\_\{p\backslash in\; P\}$ defined on {{mvar|U}} with support in {{mvar|V}} such that

$$T\; =\; \backslash sum\_\{p\; \backslash in\; P\}\; \backslash partial^p\; f\_p,$$

where the derivatives are understood in the sense of distributions. That is, for all test functions $\backslash phi$ on {{mvar|U}},

$$T\; \backslash phi\; =\; \backslash sum\_\{p\; \backslash in\; P\}\; (-1)^$$

}}

The formal definition of distributions exhibits them as a subspace of a very large space, namely the topological dual of $\backslash mathcal\{D\}(U)$ (or the Schwartz space $\backslash mathcal\{S\}(\backslash R^n)$ for tempered distributions). It is not immediately clear from the definition how exotic a distribution might be. To answer this question, it is instructive to see distributions built up from a smaller space, namely the space of continuous functions. Roughly, any distribution is locally a (multiple) derivative of a continuous function. A precise version of this result, given below, holds for distributions of compact support, tempered distributions, and general distributions. Generally speaking, no proper subset of the space of distributions contains all continuous functions and is closed under differentiation. This says that distributions are not particularly exotic objects; they are only as complicated as necessary.

{{Math theorem|name=Theorem{{sfn|Trèves|2006|pp=258-264}}|math_statement=

Let {{mvar|T}} be a distribution on {{mvar|U}}.

There exists a sequence $(T\_i)\_\{i=1\}^\backslash infty$ in $\backslash mathcal\{D\}\text{'}(U)$ such that each {{mvar|T_i}} has compact support and every compact subset $K\; \backslash subseteq\; U$ intersects the support of only finitely many $T\_i,$ and the sequence of partial sums $(S\_j)\_\{j=1\}^\backslash infty,$ defined by $S\_j\; :=\; T\_1\; +\; \backslash cdots\; +\; T\_j,$ converges in $\backslash mathcal\{D\}\text{'}(U)$ to {{mvar|T}}; in other words we have:

$$T\; =\; \backslash sum\_\{i=1\}^\backslash infty\; T\_i.$$

Recall that a sequence converges in $\backslash mathcal\{D\}\text{'}(U)$ (with its strong dual topology) if and only if it converges pointwise.

}}

By combining the above results, one may express any distribution on {{mvar|U}} as the sum of a series of distributions with compact support, where each of these distributions can in turn be written as a finite sum of distributional derivatives of continuous functions on {{mvar|U}}. In other words, for arbitrary $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(U)$ we can write:

$$T\; =\; \backslash sum\_\{i=1\}^\backslash infty\; \backslash sum\_\{p\; \backslash in\; P\_i\}\; \backslash partial^p\; f\_\{ip\},$$

where $P\_1,\; P\_2,\; \backslash ldots$ are finite sets of multi-indices and the functions $f\_\{ip\}$ are continuous.

{{Math theorem|name=Theorem{{sfn|Rudin|1991|pp=169-170}}|math_statement=

Let {{mvar|T}} be a distribution on {{mvar|U}}. For every multi-index {{mvar|p}} there exists a continuous function $g\_p$ on {{mvar|U}} such that

1. any compact subset {{mvar|K}} of {{mvar|U}} intersects the support of only finitely many $g\_p,$ and
2. $T\; =\; \backslash sum\backslash nolimits\_p\; \backslash partial^p\; g\_p.$

Moreover, if {{mvar|T}} has finite order, then one can choose $g\_p$ in such a way that only finitely many of them are non-zero.

}}

Note that the infinite sum above is well-defined as a distribution. The value of {{mvar|T}} for a given $f\; \backslash in\; \backslash mathcal\{D\}(U)$ can be computed using the finitely many $g\_\backslash alpha$ that intersect the support of $f.$

Many operations which are defined on smooth functions with compact support can also be defined for distributions. In general, if $A:\backslash mathcal\{D\}(U)\backslash to\backslash mathcal\{D\}(U)$ is a linear map that is continuous with respect to the weak topology, then it is not always possible to extend $A$ to a map $A\text{'}:\; \backslash mathcal\{D\}\text{'}(U)\backslash to\; \backslash mathcal\{D\}\text{'}(U)$ by classic extension theorems of topology or linear functional analysis.The extension theorem for mappings defined from a subspace S of a topological vector space E to the topological space E itself works for non-linear mappings as well, provided they are assumed to be uniformly continuous. But, unfortunately, this is not our case, we would desire to “extend” a linear continuous mapping A from a tvs E into another tvs F, in order to obtain a linear continuous mapping from the dual E’ to the dual F’ (note the order of spaces). In general, this is not even an extension problem, because (in general) E is not necessarily a subset of its own dual E’. Moreover, It is not a classic topological transpose problem, because the transpose of A goes from F’ to E’ and not from E’ to F’. Our case needs, indeed, a new order of ideas, involving the specific topological properties of the Laurent Schwartz spaces D(U) and D’(U), together with the fundamental concept of weak (or Schwartz) adjoint of the linear continuous operator A. The “distributional” extension of the above linear continuous operator A is possible if and only if A admits a Schwartz adjoint, that is another linear continuous operator B of the same type such that

\langle Af,g\rangle = \langle f,Bg\rangle

,

for every pair of test functions. In that condition, B is unique and the extension A’ is the transpose of the Schwartz adjoint B. {{citation needed|date=September 2020}}{{Cite book |last=Strichartz |first=Robert |title=A Guide to Distribution Theory and Fourier Transforms |year=1993 |location=USA |pages=17 |language=English}}{{clarify|date=September 2020}}

{{anchor|Transpose of a linear operator}}

{{Main|Transpose of a linear map}}

Operations on distributions and spaces of distributions are often defined using the transpose of a linear operator. This is because the transpose allows for a unified presentation of the many definitions in the theory of distributions and also because its properties are well-known in functional analysis.{{harvnb|Strichartz|1994|loc=§2.3}}; {{harvnb|Trèves|2006}}. For instance, the well-known Hermitian adjoint of a linear operator between Hilbert spaces is just the operator's transpose (but with the Riesz representation theorem used to identify each Hilbert space with its continuous dual space). In general, the transpose of a continuous linear map $A\; :\; X\; \backslash to\; Y$ is the linear map

$$\{\}^\{t\}A\; :\; Y\text{'}\; \backslash to\; X\text{'}\; \backslash qquad\; \backslash text\{\; defined\; by\; \}\; \backslash qquad\; \{\}^\{t\}A(y\text{'})\; :=\; y\text{'}\; \backslash circ\; A,$$

or equivalently, it is the unique map satisfying $\backslash langle\; y\text{'},\; A(x)\backslash rangle\; =\; \backslash left\backslash langle\; \{\}^\{t\}A\; (y\text{'}),\; x\; \backslash right\backslash rangle$ for all $x\; \backslash in\; X$ and all $y\text{'}\; \backslash in\; Y\text{'}$ (the prime symbol in $y\text{'}$ does not denote a derivative of any kind; it merely indicates that $y\text{'}$ is an element of the continuous dual space $Y\text{'}$). Since $A$ is continuous, the transpose $\{\}^\{t\}A\; :\; Y\text{'}\; \backslash to\; X\text{'}$ is also continuous when both duals are endowed with their respective strong dual topologies; it is also continuous when both duals are endowed with their respective weak* topologies (see the articles polar topology and dual system for more details).

In the context of distributions, the characterization of the transpose can be refined slightly. Let $A\; :\; \backslash mathcal\{D\}(U)\; \backslash to\; \backslash mathcal\{D\}(U)$ be a continuous linear map. Then by definition, the transpose of $A$ is the unique linear operator $\{\}^tA\; :\; \backslash mathcal\{D\}\text{'}(U)\; \backslash to\; \backslash mathcal\{D\}\text{'}(U)$ that satisfies:

$$\backslash langle\; \{\}^\{t\}A(T),\; \backslash phi\; \backslash rangle\; =\; \backslash langle\; T,\; A(\backslash phi)\; \backslash rangle\; \backslash quad\; \backslash text\{\; for\; all\; \}\; \backslash phi\; \backslash in\; \backslash mathcal\{D\}(U)\; \backslash text\{\; and\; all\; \}\; T\; \backslash in\; \backslash mathcal\{D\}\text{'}(U).$$

Since $\backslash mathcal\{D\}(U)$ is dense in $\backslash mathcal\{D\}\text{'}(U)$ (here, $\backslash mathcal\{D\}(U)$ actually refers to the set of distributions $\backslash left\backslash \{D\_\backslash psi\; :\; \backslash psi\; \backslash in\; \backslash mathcal\{D\}(U)\backslash right\backslash \}$) it is sufficient that the defining equality hold for all distributions of the form $T\; =\; D\_\backslash psi$ where $\backslash psi\; \backslash in\; \backslash mathcal\{D\}(U).$ Explicitly, this means that a continuous linear map $B\; :\; \backslash mathcal\{D\}\text{'}(U)\; \backslash to\; \backslash mathcal\{D\}\text{'}(U)$ is equal to $\{\}^\{t\}A$ if and only if the condition below holds:

$$\backslash langle\; B(D\_\backslash psi),\; \backslash phi\; \backslash rangle\; =\; \backslash langle\; \{\}^\{t\}A(D\_\backslash psi),\; \backslash phi\; \backslash rangle\; \backslash quad\; \backslash text\{\; for\; all\; \}\; \backslash phi,\; \backslash psi\; \backslash in\; \backslash mathcal\{D\}(U)$$

where the right-hand side equals $\backslash langle\; \{\}^\{t\}A(D\_\backslash psi),\; \backslash phi\; \backslash rangle\; =\; \backslash langle\; D\_\backslash psi,\; A(\backslash phi)\; \backslash rangle\; =\; \backslash langle\; \backslash psi,\; A(\backslash phi)\; \backslash rangle\; =\; \backslash int\_U\; \backslash psi\; \backslash cdot\; A(\backslash phi)\; \backslash ,dx.$

Let $A\; :\; \backslash mathcal\{D\}(U)\; \backslash to\; \backslash mathcal\{D\}(U)$ be the partial derivative operator $\backslash tfrac\{\backslash partial\}\{\backslash partial\; x\_k\}.$ To extend $A$ we compute its transpose:

$$\backslash begin\{align\}$$

\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}

Therefore $\{\}^\{t\}A\; =\; -A.$ Thus, the partial derivative of $T$ with respect to the coordinate $x\_k$ is defined by the formula

$$\backslash left\backslash langle\; \backslash frac\{\backslash partial\; T\}\{\backslash partial\; x\_k\},\; \backslash phi\; \backslash right\backslash rangle\; =\; -\; \backslash left\backslash langle\; T,\; \backslash frac\{\backslash partial\; \backslash phi\}\{\backslash partial\; x\_k\}\; \backslash right\backslash rangle\; \backslash qquad\; \backslash text\{\; for\; all\; \}\; \backslash phi\; \backslash in\; \backslash mathcal\{D\}(U).$$

With this definition, every distribution is infinitely differentiable, and the derivative in the direction $x\_k$ is a linear operator on $\backslash mathcal\{D\}\text{'}(U).$

More generally, if $\backslash alpha$ is an arbitrary multi-index, then the partial derivative $\backslash partial^\backslash alpha\; T$ of the distribution $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(U)$ is defined by

$$\backslash langle\; \backslash partial^\backslash alpha\; T,\; \backslash phi\; \backslash rangle\; =\; (-1)^$$

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

If $T$ is a distribution in $\backslash R$ then

$$\backslash lim\_\{x\; \backslash to\; 0\}\; \backslash frac\{T\; -\; \backslash tau\_x\; T\}\{x\}\; =\; T\text{'}\backslash in\; \backslash mathcal\{D\}\text{'}(\backslash R),$$

where $T\text{'}$ is the derivative of $T$ and $\backslash tau\_x$ is a translation by $x;$ thus the derivative of $T$ may be viewed as a limit of quotients.{{sfn|Rudin|1991|p=180}}

A linear differential operator in $U$ with smooth coefficients acts on the space of smooth functions on $U.$ Given such an operator

$P\; :=\; \backslash sum\_\backslash alpha\; c\_\backslash alpha\; \backslash partial^\backslash alpha,$

we would like to define a continuous linear map, $D\_P$ that extends the action of $P$ on $C^\backslash infty(U)$ to distributions on $U.$ In other words, we would like to define $D\_P$ such that the following diagram commutes:

$$\backslash begin\{matrix\}$$

\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}

where the vertical maps are given by assigning $f\; \backslash in\; C^\backslash infty(U)$ its canonical distribution $D\_f\; \backslash in\; \backslash mathcal\{D\}\text{'}(U),$ which is defined by:

$$D\_f(\backslash phi)\; =\; \backslash langle\; f,\; \backslash phi\; \backslash rangle\; :=\; \backslash int\_U\; f(x)\; \backslash phi(x)\; \backslash ,dx\; \backslash quad\; \backslash text\{\; for\; all\; \}\; \backslash phi\; \backslash in\; \backslash mathcal\{D\}(U).$$

With this notation, the diagram commuting is equivalent to:

$$D\_\{P(f)\}\; =\; D\_PD\_f\; \backslash qquad\; \backslash text\{\; for\; all\; \}\; f\; \backslash in\; C^\backslash infty(U).$$

To find $D\_P,$ the transpose $\{\}^\{t\}\; P\; :\; \backslash mathcal\{D\}\text{'}(U)\; \backslash to\; \backslash mathcal\{D\}\text{'}(U)$ of the continuous induced map $P\; :\; \backslash mathcal\{D\}(U)\backslash to\; \backslash mathcal\{D\}(U)$ defined by $\backslash phi\; \backslash mapsto\; P(\backslash phi)$ is considered in the lemma below.

This leads to the following definition of the differential operator on $U$ called {{em|the formal transpose of $P,$}} which will be denoted by $P\_*$ to avoid confusion with the transpose map, that is defined by

$$P\_*\; :=\; \backslash sum\_\backslash alpha\; b\_\backslash alpha\; \backslash partial^\backslash alpha\; \backslash quad\; \backslash text\{\; where\; \}\; \backslash quad\; b\_\backslash alpha\; :=\; \backslash sum\_\{\backslash beta\; \backslash geq\; \backslash alpha\}\; (-1)^$$

{{math theorem|name=Lemma|math_statement= Let $P$ be a linear differential operator with smooth coefficients in $U.$ Then for all $\backslash phi\; \backslash in\; \backslash mathcal\{D\}(U)$ we have

$$\backslash left\backslash langle\; \{\}^\{t\}P(D\_f),\; \backslash phi\; \backslash right\backslash rangle\; =\; \backslash left\backslash langle\; D\_\{P\_*(f)\},\; \backslash phi\; \backslash right\backslash rangle,$$

which is equivalent to:

$$\{\}^\{t\}P(D\_f)\; =\; D\_\{P\_*(f)\}.$$}}

{{collapse top|title=Proof|left=true}}

As discussed above, for any $\backslash phi\; \backslash in\; \backslash mathcal\{D\}(U),$ the transpose may be calculated by:

$$\backslash begin\{align\}$$

\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)^

\end{align}

For the last line we used integration by parts combined with the fact that $\backslash phi$ and therefore all the functions $f\; (x)c\_\backslash alpha\; (x)\; \backslash partial^\backslash alpha\; \backslash phi(x)$ have compact support.For example, let $U\; =\; \backslash R$ and take $P$ to be the ordinary derivative for functions of one real variable and assume the support of $\backslash phi$ to be contained in the finite interval $(a,b),$ then since $\backslash operatorname\{supp\}(\backslash phi)\; \backslash subseteq\; (a,\; b)$

$$\backslash begin\{align\}$$

\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}

where the last equality is because $\backslash phi(a)\; =\; \backslash phi(b)\; =\; 0.$ Continuing the calculation above, for all $\backslash phi\; \backslash in\; \backslash mathcal\{D\}(U):$

$$\backslash begin\{align\}$$

\left\langle {}^{t}P(D_f), \phi \right\rangle &=\sum\nolimits_\alpha (-1)^

&= \int_U \phi(x) \sum\nolimits_\alpha (-1)^

&= \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)^

&= \int_U \phi(x) \left[ \sum_\alpha \left[ \sum_{\beta \geq \alpha} (-1)^

&= \int_U \phi(x) \left[\sum\nolimits_\alpha b_\alpha(x) (\partial^\alpha f)(x) \right] \, dx && b_\alpha:=\sum_{\beta \geq \alpha} (-1)^

&= \left\langle \left(\sum\nolimits_\alpha b_\alpha \partial^\alpha \right) (f), \phi \right\rangle

\end{align}

{{collapse bottom}}

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

$$\backslash begin\{align\}$$

\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}

We call the continuous linear operator $D\_P\; :=\; \{\}^\{t\}P\_*\; :\; \backslash mathcal\{D\}\text{'}(U)\; \backslash to\; \backslash mathcal\{D\}\text{'}(U)$ the {{em|differential operator on distributions extending $P$}}.{{sfn|Trèves|2006|pp=247-252}} Its action on an arbitrary distribution $S$ is defined via:

$$D\_P(S)(\backslash phi)\; =\; S\backslash left(P\_*(\backslash phi)\backslash right)\; \backslash quad\; \backslash text\{\; for\; all\; \}\; \backslash phi\; \backslash in\; \backslash mathcal\{D\}(U).$$

If $(T\_i)\_\{i=1\}^\backslash infty$ converges to $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(U)$ then for every multi-index $\backslash alpha,\; (\backslash partial^\backslash alpha\; T\_i)\_\{i=1\}^\backslash infty$ converges to $\backslash partial^\backslash alpha\; T\; \backslash in\; \backslash mathcal\{D\}\text{'}(U).$

A differential operator of order 0 is just multiplication by a smooth function. And conversely, if $f$ is a smooth function then $P\; :=\; f(x)$ is a differential operator of order 0, whose formal transpose is itself (that is, $P\_*\; =\; P$). The induced differential operator $D\_P\; :\; \backslash mathcal\{D\}\text{'}(U)\; \backslash to\; \backslash mathcal\{D\}\text{'}(U)$ maps a distribution $T$ to a distribution denoted by $fT\; :=\; D\_P(T).$ 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 $T$ on $U$ by a smooth function $m\; :\; U\; \backslash to\; \backslash R.$ The product $mT$ is defined by

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

This definition coincides with the transpose definition since if $M\; :\; \backslash mathcal\{D\}(U)\; \backslash to\; \backslash mathcal\{D\}(U)$ is the operator of multiplication by the function $m$ (that is, $(M\backslash phi)(x)\; =\; m(x)\backslash phi(x)$), then

$$\backslash int\_U\; (M\; \backslash phi)(x)\; \backslash psi(x)\backslash ,dx\; =\; \backslash int\_U\; m(x)\; \backslash phi(x)\; \backslash psi(x)\backslash ,d\; x\; =\; \backslash int\_U\; \backslash phi(x)\; m(x)\; \backslash psi(x)\; \backslash ,d\; x\; =\; \backslash int\_U\; \backslash phi(x)\; (M\; \backslash psi)(x)\backslash ,d\; x,$$

so that $\{\}^tM\; =\; M.$

Under multiplication by smooth functions, $\backslash mathcal\{D\}\text{'}(U)$ is a module over the ring $C^\backslash infty(U).$ 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 $\backslash delta$ is the Dirac delta distribution on $\backslash R,$ then $m\; \backslash delta\; =\; m(0)\; \backslash delta,$ and if $\backslash delta^\text{'}$ is the derivative of the delta distribution, then

$$m\backslash delta\text{'}\; =\; m(0)\; \backslash delta\text{'}\; -\; m\text{'}\; \backslash delta\; =\; m(0)\; \backslash delta\text{'}\; -\; m\text{'}(0)\; \backslash delta.$$

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

Example. The product of any distribution $T$ with the function that is identically {{math|1}} on $U$ is equal to $T.$

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

If $(T\_i)\_\{i=1\}^\backslash infty$ converges to $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(U)$ and $(f\_i)\_\{i=1\}^\backslash infty$ converges to $f\; \backslash in\; C^\backslash infty(U)$ then $(f\_i\; T\_i)\_\{i=1\}^\backslash infty$ converges to $fT\; \backslash in\; \backslash mathcal\{D\}\text{'}(U).$

It is easy to define the product of a distribution with a smooth function, or more generally the product of two distributions whose singular supports are disjoint.{{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)}} With more effort, it is possible to define a well-behaved product of several distributions provided their wave front sets 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 $\backslash operatorname\{p.v.\}\; \backslash frac\{1\}\{x\}$ is the distribution obtained by the Cauchy principal value

$$\backslash left(\backslash operatorname\{p.v.\}\; \backslash frac\{1\}\{x\}\backslash right)(\backslash phi)\; =\; \backslash lim\_\{\backslash varepsilon\backslash to\; 0^+\}\; \backslash int\_\{|x|\; \backslash geq\; \backslash varepsilon\}\; \backslash frac\{\backslash phi(x)\}\{x\}\backslash ,\; dx\; \backslash quad\; \backslash text\{\; for\; all\; \}\; \backslash phi\; \backslash in\; \backslash mathcal\{S\}(\backslash R).$$

If $\backslash delta$ is the Dirac delta distribution then

$$(\backslash delta\; \backslash times\; x)\; \backslash times\; \backslash operatorname\{p.v.\}\; \backslash frac\{1\}\{x\}\; =\; 0$$

but,

$$\backslash delta\; \backslash times\; \backslash left(x\; \backslash times\; \backslash operatorname\{p.v.\}\; \backslash frac\{1\}\{x\}\backslash right)\; =\; \backslash delta$$

so the product of a distribution by a smooth function (which is always well-defined) cannot be extended to an 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 of 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 algebras of generalized functions have been developed, among which Colombeau's (simplified) algebra is maybe the most popular in use today.

Inspired by Lyons' rough path theory,{{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}} Martin Hairer proposed a consistent way of multiplying distributions with certain structures (regularity structures{{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 }}), available in many examples from stochastic analysis, notably stochastic partial differential equations. See also Gubinelli–Imkeller–Perkowski (2015) for a related development based on Bony's paraproduct from Fourier analysis.

Let $T$ be a distribution on $U.$ Let $V$ be an open set in $\backslash R^n$ and $F\; :\; V\; \backslash to\; U.$ If $F$ is a submersion then it is possible to define

$$T\; \backslash circ\; F\; \backslash in\; \backslash mathcal\{D\}\text{'}(V).$$

This is {{em|the composition of the distribution $T$ with $F$}}, and is also called {{em|the pullback of $T$ along $F$}}, sometimes written

$$F^\backslash sharp\; :\; T\; \backslash mapsto\; F^\backslash sharp\; T\; =\; T\; \backslash circ\; F.$$

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 $d\; F(x)$ of $F$ is a surjective linear map for every $x\; \backslash in\; V.$ A necessary (but not sufficient) condition for extending $F^\{\backslash \#\}$ to distributions is that $F$ be an open mapping.See for example {{harvnb|Hörmander|1983|loc=Theorem 6.1.1}}. The Inverse function theorem ensures that a submersion satisfies this condition.

If $F$ is a submersion, then $F^\{\backslash \#\}$ is defined on distributions by finding the transpose map. The uniqueness of this extension is guaranteed since $F^\{\backslash \#\}$ is a continuous linear operator on $\backslash mathcal\{D\}(U).$ Existence, however, requires using the change of variables formula, the inverse function theorem (locally), and a partition of unity argument.See {{harvnb|Hörmander|1983|loc=Theorem 6.1.2}}.

In the special case when $F$ is a diffeomorphism from an open subset $V$ of $\backslash R^n$ onto an open subset $U$ of $\backslash R^n$ change of variables under the integral gives:

$$\backslash int\_V\; \backslash phi\backslash circ\; F(x)\; \backslash psi(x)\backslash ,dx\; =\; \backslash int\_U\; \backslash phi(x)\; \backslash psi\; \backslash left(F^\{-1\}(x)\; \backslash right)\; \backslash left|\backslash det\; dF^\{-1\}(x)\; \backslash right|\backslash ,dx.$$

In this particular case, then, $F^\{\backslash \#\}$ is defined by the transpose formula:

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

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 $\backslash R^n$ then we denote by $f\backslash ast\; g$ {{em|the convolution of $f$ and $g,$}} defined at $x\; \backslash in\; \backslash R^n$ to be the integral

$$(f\; \backslash ast\; g)(x)\; :=\; \backslash int\_\{\backslash R^n\}\; f(x-y)\; g(y)\; \backslash ,dy\; =\; \backslash int\_\{\backslash R^n\}\; f(y)g(x-y)\; \backslash ,dy$$

provided that the integral exists. If $1\; \backslash leq\; p,\; q,\; r\; \backslash leq\; \backslash infty$ are such that $\backslash frac\{1\}\{r\}\; =\; \backslash frac\{1\}\{p\}\; +\; \backslash frac\{1\}\{q\}\; -\; 1$ then for any functions $f\; \backslash in\; L^p(\backslash R^n)$ and $g\; \backslash in\; L^q(\backslash R^n)$ we have $f\; \backslash ast\; g\; \backslash in\; L^r(\backslash R^n)$ and $\backslash |f\backslash ast\; g\backslash |\_\{L^r\}\; \backslash leq\; \backslash |f\backslash |\_\{L^p\}\; \backslash |g\backslash |\_\{L^q\}.${{sfn|Trèves|2006|pp=278-283}} If $f$ and $g$ are continuous functions on $\backslash R^n,$ at least one of which has compact support, then $\backslash operatorname\{supp\}(f\; \backslash ast\; g)\; \backslash subseteq\; \backslash operatorname\{supp\}\; (f)\; +\; \backslash operatorname\{supp\}\; (g)$ and if $A\backslash subseteq\; \backslash R^n$ then the value of $f\backslash ast\; g$ on $A$ do {{em|not}} depend on the values of $f$ outside of the Minkowski sum $A\; -\backslash operatorname\{supp\}\; (g)\; =\; \backslash \{a-s\; :\; a\backslash in\; A,\; s\backslash in\; \backslash operatorname\{supp\}(g)\backslash \}.${{sfn|Trèves|2006|pp=278-283}}

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

Given $a\; \backslash in\; \backslash R^n,$ the translation operator $\backslash tau\_a$ sends $f\; :\; \backslash R^n\; \backslash to\; \backslash Complex$ to $\backslash tau\_a\; f\; :\; \backslash R^n\; \backslash to\; \backslash Complex,$ defined by $\backslash tau\_a\; f(y)\; =\; f(y-a).$ This can be extended by the transpose to distributions in the following way: given a distribution $T,$ {{em|the translation of $T$ by $a$}} is the distribution $\backslash tau\_a\; T\; :\; \backslash mathcal\{D\}(\backslash R^n)\; \backslash to\; \backslash Complex$ defined by $\backslash tau\_a\; T(\backslash phi)\; :=\; \backslash left\backslash langle\; T,\; \backslash tau\_\{-a\}\; \backslash phi\; \backslash right\backslash rangle.${{sfn|Trèves|2006|pp=284-297}}See for example {{harvnb|Rudin|1991|loc=§6.29}}.

Given $f\; :\; \backslash R^n\; \backslash to\; \backslash Complex,$ define the function $\backslash tilde\{f\}\; :\; \backslash R^n\; \backslash to\; \backslash Complex$ by $\backslash tilde\{f\}(x)\; :=\; f(-x).$ Given a distribution $T,$ let $\backslash tilde\{T\}\; :\; \backslash mathcal\{D\}(\backslash R^n)\; \backslash to\; \backslash Complex$ be the distribution defined by $\backslash tilde\{T\}(\backslash phi)\; :=\; T\; \backslash left(\backslash tilde\{\backslash phi\}\backslash right).$ The operator $T\; \backslash mapsto\; \backslash tilde\{T\}$ is called {{em|the symmetry with respect to the origin}}.{{sfn|Trèves|2006|pp=284-297}}

Convolution with $f\; \backslash in\; \backslash mathcal\{D\}(\backslash R^n)$ defines a linear map:

$$\backslash begin\{alignat\}\{4\}$$

C_f : \,& \mathcal{D}(\R^n) && \to \,&& \mathcal{D}(\R^n) \\

& g && \mapsto\,&& f \ast g \\

\end{alignat}

which is continuous with respect to the canonical LF space topology on $\backslash mathcal\{D\}(\backslash R^n).$

Convolution of $f$ with a distribution $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(\backslash R^n)$ can be defined by taking the transpose of $C\_f$ relative to the duality pairing of $\backslash mathcal\{D\}(\backslash R^n)$ with the space $\backslash mathcal\{D\}\text{'}(\backslash R^n)$ of distributions.{{sfn|Trèves|2006|loc=Chapter 27}} If $f,\; g,\; \backslash phi\; \backslash in\; \backslash mathcal\{D\}(\backslash R^n),$ then by Fubini's theorem

$$\backslash langle\; C\_fg,\; \backslash phi\; \backslash rangle\; =\; \backslash int\_\{\backslash R^n\}\backslash phi(x)\backslash int\_\{\backslash R^n\}f(x-y)\; g(y)\; \backslash ,dy\; \backslash ,dx\; =\; \backslash left\backslash langle\; g,C\_\{\backslash tilde\{f\}\}\backslash phi\; \backslash right\backslash rangle.$$

Extending by continuity, the convolution of $f$ with a distribution $T$ is defined by

$$\backslash langle\; f\; \backslash ast\; T,\; \backslash phi\; \backslash rangle\; =\; \backslash left\backslash langle\; T,\; \backslash tilde\{f\}\; \backslash ast\; \backslash phi\; \backslash right\backslash rangle,\; \backslash quad\; \backslash text\{\; for\; all\; \}\; \backslash phi\; \backslash in\; \backslash mathcal\{D\}(\backslash R^n).$$

An alternative way to define the convolution of a test function $f$ and a distribution $T$ is to use the translation operator $\backslash tau\_a.$ The convolution of the compactly supported function $f$ and the distribution $T$ is then the function defined for each $x\; \backslash in\; \backslash R^n$ by

$$(f\; \backslash ast\; T)(x)\; =\; \backslash left\backslash langle\; T,\; \backslash tau\_x\; \backslash tilde\{f\}\; \backslash right\backslash rangle.$$

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, and if $f$ is a polynomial (resp. an exponential function, an analytic function, the restriction of an entire analytic function on $\backslash Complex^n$ to $\backslash R^n,$ the restriction of an entire function of exponential type in $\backslash Complex^n$ to $\backslash R^n$), then the same is true of $T\; \backslash ast\; f.${{sfn|Trèves|2006|pp=284-297}} If the distribution $T$ has compact support as well, then $f\backslash ast\; T$ is a compactly supported function, and the Titchmarsh convolution theorem {{harvtxt|Hörmander|1983|loc=Theorem 4.3.3}} implies that:

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

where $\backslash operatorname\{ch\}$ denotes the convex hull and $\backslash operatorname\{supp\}$ denotes the support.

Let $f\; \backslash in\; C^\backslash infty(\backslash R^n)$ and $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(\backslash R^n)$ and assume that at least one of $f$ and $T$ has compact support. The {{em|convolution}} of $f$ and $T,$ denoted by $f\; \backslash ast\; T$ or by $T\; \backslash ast\; f,$ is the smooth function:{{sfn|Trèves|2006|pp=284-297}}

$$\backslash begin\{alignat\}\{4\}$$

f \ast T : \,& \R^n && \to \,&& \Complex \\

& x && \mapsto\,&& \left\langle T, \tau_x \tilde{f} \right\rangle \\

\end{alignat}

satisfying for all $p\; \backslash in\; \backslash N^n$:

$$\backslash begin\{align\}$$

&\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}

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

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

Example.{{sfn|Rudin|1991|pp=149-181}} Let $H$ be the Heaviside function on $\backslash R.$ For any $\backslash phi\; \backslash in\; \backslash mathcal\{D\}(\backslash R),$

$$(H\; \backslash ast\; \backslash phi)(x)\; =\; \backslash int\_\{-\backslash infty\}^x\; \backslash phi(t)\; \backslash ,\; dt.$$

Let $\backslash delta$ be the Dirac measure at 0 and let $\backslash delta\text{'}$ be its derivative as a distribution. Then $\backslash delta\text{'}\; \backslash ast\; H\; =\; \backslash delta$ and $1\; \backslash ast\; \backslash delta\text{'}\; =\; 0.$ Importantly, the associative law fails to hold:

$$1\; =\; 1\; \backslash ast\; \backslash delta\; =\; 1\; \backslash ast\; (\backslash delta\text{'}\; \backslash ast\; H\; )\; \backslash neq\; (1\; \backslash ast\; \backslash delta\text{'})\; \backslash ast\; H\; =\; 0\; \backslash ast\; H\; =\; 0.$$

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

$$S\; \backslash ast\; (T\; \backslash ast\; \backslash phi)\; =\; (S\; \backslash ast\; T)\; \backslash ast\; \backslash phi$$

continues to hold for all test functions $\backslash phi.${{harvnb|Hörmander|1983|loc=§IV.2}} proves the uniqueness of such an extension.

It is also possible to provide a more explicit characterization of the convolution of distributions.{{sfn|Trèves|2006|loc=Chapter 27}} Suppose that $S$ and $T$ are distributions and that $S$ has compact support. Then the linear maps

$$\backslash begin\{alignat\}\{9\}$$

\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}

are continuous. The transposes of these maps:

$$\{\}^\{t\}\backslash left(\backslash bullet\; \backslash ast\; \backslash tilde\{S\}\backslash right)\; :\; \backslash mathcal\{D\}\text{'}(\backslash R^n)\; \backslash to\; \backslash mathcal\{D\}\text{'}(\backslash R^n)\; \backslash qquad\; \{\}^\{t\}\backslash left(\backslash bullet\; \backslash ast\; \backslash tilde\{T\}\backslash right)\; :\; \backslash mathcal\{E\}\text{'}(\backslash R^n)\; \backslash to\; \backslash mathcal\{D\}\text{'}(\backslash R^n)$$

are consequently continuous and it can also be shown that{{sfn|Trèves|2006|pp=284-297}}

$$\{\}^\{t\}\backslash left(\backslash bullet\; \backslash ast\; \backslash tilde\{S\}\backslash right)(T)\; =\; \{\}^\{t\}\backslash left(\backslash bullet\; \backslash ast\; \backslash tilde\{T\}\backslash right)(S).$$

This common value is called {{em|the convolution of $S$ and $T$}} and it is a distribution that is denoted by $S\; \backslash ast\; T$ or $T\; \backslash ast\; S.$ It satisfies $\backslash operatorname\{supp\}\; (S\; \backslash ast\; T)\; \backslash subseteq\; \backslash operatorname\{supp\}(S)\; +\; \backslash operatorname\{supp\}(T).${{sfn|Trèves|2006|pp=284-297}} If $S$ and $T$ are two distributions, at least one of which has compact support, then for any $a\; \backslash in\; \backslash R^n,$ $\backslash tau\_a(S\; \backslash ast\; T)\; =\; \backslash left(\backslash tau\_a\; S\backslash right)\; \backslash ast\; T\; =\; S\; \backslash ast\; \backslash left(\backslash tau\_a\; T\backslash right).${{sfn|Trèves|2006|pp=284-297}} If $T$ is a distribution in $\backslash R^n$ and if $\backslash delta$ is a Dirac measure then $T\; \backslash ast\; \backslash delta\; =\; T\; =\; \backslash delta\; \backslash ast\; T$;{{sfn|Trèves|2006|pp=284-297}} thus $\backslash delta$ is the identity element of the convolution operation. Moreover, if $f$ is a function then $f\; \backslash ast\; \backslash delta^\{\backslash prime\}\; =\; f^\{\backslash prime\}\; =\; \backslash delta^\{\backslash prime\}\; \backslash ast\; f$ where now the associativity of convolution implies that $f^\{\backslash prime\}\; \backslash ast\; g\; =\; g^\{\backslash prime\}\; \backslash ast\; f$ for all functions $f$ and $g.$

Suppose that it is $T$ that has compact support. For $\backslash phi\; \backslash in\; \backslash mathcal\{D\}(\backslash R^n)$ consider the function

$$\backslash psi(x)\; =\; \backslash langle\; T,\; \backslash tau\_\{-x\}\; \backslash phi\; \backslash rangle.$$

It can be readily shown that this defines a smooth function of $x,$ which moreover has compact support. The convolution of $S$ and $T$ is defined by

$$\backslash langle\; S\; \backslash ast\; T,\; \backslash phi\; \backslash rangle\; =\; \backslash langle\; S,\; \backslash psi\; \backslash rangle.$$

This generalizes the classical notion of convolution of functions and is compatible with differentiation in the following sense: for every multi-index $\backslash alpha.$

$$\backslash partial^\backslash alpha(S\; \backslash ast\; T)\; =\; (\backslash partial^\backslash alpha\; S)\; \backslash ast\; T\; =\; S\; \backslash ast\; (\backslash partial^\backslash alpha\; T).$$

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 $S$ and $T.$See for instance {{harvnb|Gel'fand|Shilov|1966–1968|loc=v. 1, pp. 103–104}} and {{harvnb|Benedetto|1997|loc=Definition 2.5.8}}.

The convolution of distributions with compact support induces a continuous bilinear map $\backslash mathcal\{E\}\text{'}\; \backslash times\; \backslash mathcal\{E\}\text{'}\; \backslash to\; \backslash mathcal\{E\}\text{'}$ defined by $(S,T)\; \backslash mapsto\; S\; *\; T,$ where $\backslash mathcal\{E\}\text{'}$ denotes the space of distributions with compact support.{{sfn|Trèves|2006|p=423}} However, the convolution map as a function $\backslash mathcal\{E\}\text{'}\; \backslash times\; \backslash mathcal\{D\}\text{'}\; \backslash to\; \backslash mathcal\{D\}\text{'}$ is {{em|not}} continuous{{sfn|Trèves|2006|p=423}} although it is separately continuous.{{sfn|Trèves|2006|p=294}} The convolution maps $\backslash mathcal\{D\}(\backslash R^n)\; \backslash times\; \backslash mathcal\{D\}\text{'}\; \backslash to\; \backslash mathcal\{D\}\text{'}$ and $\backslash mathcal\{D\}(\backslash R^n)\; \backslash times\; \backslash mathcal\{D\}\text{'}\; \backslash to\; \backslash mathcal\{D\}(\backslash R^n)$ given by $(f,\; T)\; \backslash mapsto\; f\; *\; T$ 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}}

In general, regularity is required for multiplication products, and locality is required for convolution products. It is expressed in the following extension of the Convolution Theorem which guarantees the existence of both convolution and multiplication products. Let $F(\backslash alpha)\; =\; f\; \backslash in\; \backslash mathcal\{O\}\text{'}\_C$ be a rapidly decreasing tempered distribution or, equivalently, $F(f)\; =\; \backslash alpha\; \backslash in\; \backslash mathcal\{O\}\_M$ be an ordinary (slowly growing, smooth) function within the space of tempered distributions and let $F$ be the normalized (unitary, ordinary frequency) Fourier transform.{{cite book|last=Folland|first=G.B.|title=Harmonic Analysis in Phase Space|publisher=Princeton University Press|publication-place=Princeton, NJ|year=1989}} Then, according to {{harvtxt|Schwartz|1951}},

$$F(f\; *\; g)\; =\; F(f)\; \backslash cdot\; F(g)\; \backslash qquad\; \backslash text\{\; and\; \}\; \backslash qquad\; F(\backslash alpha\; \backslash cdot\; g)\; =\; F(\backslash alpha)\; *\; F(g)$$

hold within the space of tempered distributions.{{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}}{{cite book|last=Barros-Neto|first=José|title=An Introduction to the Theory of Distributions|publisher=Dekker|publication-place=New York, NY|year=1973}}{{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}} In particular, these equations become the Poisson Summation Formula if $g\; \backslash equiv\; \backslash operatorname\{\backslash text\{Ш\}\}$ is the Dirac Comb.{{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}} The space of all rapidly decreasing tempered distributions is also called the space of {{em|convolution operators}} $\backslash mathcal\{O\}\text{'}\_C$ and the space of all ordinary functions within the space of tempered distributions is also called the space of {{em|multiplication operators}} $\backslash mathcal\{O\}\_M.$ More generally, $F(\backslash mathcal\{O\}\text{'}\_C)\; =\; \backslash mathcal\{O\}\_M$ and $F(\backslash mathcal\{O\}\_M)\; =\; \backslash mathcal\{O\}\text{'}\_C.${{sfn|Trèves|2006|pp=318-319}}{{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}} A particular case is the Paley-Wiener-Schwartz Theorem which states that $F(\backslash mathcal\{E\}\text{'})\; =\; \backslash operatorname\{PW\}$ and $F(\backslash operatorname\{PW\}\; )\; =\; \backslash mathcal\{E\}\text{'}.$ This is because $\backslash mathcal\{E\}\text{'}\; \backslash subseteq\; \backslash mathcal\{O\}\text{'}\_C$ and $\backslash operatorname\{PW\}\; \backslash subseteq\; \backslash mathcal\{O\}\_M.$ In other words, compactly supported tempered distributions $\backslash mathcal\{E\}\text{'}$ belong to the space of {{em|convolution operators}} $\backslash mathcal\{O\}\text{'}\_C$ and

Paley-Wiener functions $\backslash operatorname\{PW\},$ better known as bandlimited functions, belong to the space of {{em|multiplication operators}} $\backslash mathcal\{O\}\_M.${{sfn|Schwartz|1951}}

For example, let $g\; \backslash equiv\; \backslash operatorname\{\backslash text\{Ш\}\}\; \backslash in\; \backslash mathcal\{S\}\text{'}$ be the Dirac comb and $f\; \backslash equiv\; \backslash delta\; \backslash in\; \backslash mathcal\{E\}\text{'}$ be the Dirac delta;then $\backslash alpha\; \backslash equiv\; 1\; \backslash in\; \backslash operatorname\{PW\}$ is the function that is constantly one and both equations yield the Dirac-comb identity. Another example is to let $g$ be the Dirac comb and $f\; \backslash equiv\; \backslash operatorname\{rect\}\; \backslash in\; \backslash mathcal\{E\}\text{'}$ be the rectangular function; then $\backslash alpha\; \backslash equiv\; \backslash operatorname\{sinc\}\; \backslash in\; \backslash operatorname\{PW\}$ is the sinc function and both equations yield the Classical Sampling Theorem for suitable $\backslash operatorname\{rect\}$ functions. More generally, if $g$ is the Dirac comb and $f\; \backslash in\; \backslash mathcal\{S\}\; \backslash subseteq\; \backslash mathcal\{O\}\text{'}\_C\; \backslash cap\; \backslash mathcal\{O\}\_M$ is a smooth window function (Schwartz function), for example, the Gaussian, then $\backslash alpha\; \backslash in\; \backslash mathcal\{S\}$ is another smooth window function (Schwartz function). They are known as mollifiers, especially in partial differential equations theory, or as regularizers in physics because they allow turning generalized functions into regular functions.

Let $U\; \backslash subseteq\; \backslash R^m$ and $V\; \backslash subseteq\; \backslash R^n$ be open sets. Assume all vector spaces to be over the field $\backslash mathbb\{F\},$ where $\backslash mathbb\{F\}=\backslash R$ or $\backslash Complex.$ For $f\; \backslash in\; \backslash mathcal\{D\}(U\; \backslash times\; V)$ define for every $u\; \backslash in\; U$ and every $v\; \backslash in\; V$ the following functions:

$$\backslash begin\{alignat\}\{9\}$$

f_u : \,& V && \to \,&& \mathbb{F} && \quad \text{ and } \quad && f^v : \,&& U && \to \,&& \mathbb{F} \\

& y && \mapsto\,&& f(u, y) && && && x && \mapsto\,&& f(x, v) \\

\end{alignat}

Given $S\; \backslash in\; \backslash mathcal\{D\}^\{\backslash prime\}(U)$ and $T\; \backslash in\; \backslash mathcal\{D\}^\{\backslash prime\}(V),$ define the following functions:

$$\backslash begin\{alignat\}\{9\}$$

\langle S, f^{\bullet}\rangle : \,& V && \to \,&& \mathbb{F} && \quad \text{ and } \quad && \langle T, f_{\bullet}\rangle : \,&& U && \to \,&& \mathbb{F} \\

& v && \mapsto\,&& \langle S, f^v \rangle && && && u && \mapsto\,&& \langle T, f_u \rangle \\

\end{alignat}

where $\backslash langle\; T,\; f\_\{\backslash bullet\}\backslash rangle\; \backslash in\; \backslash mathcal\{D\}(U)$ and $\backslash langle\; S,\; f^\{\backslash bullet\}\backslash rangle\; \backslash in\; \backslash mathcal\{D\}(V).$

These definitions associate every $S\; \backslash in\; \backslash mathcal\{D\}\text{'}(U)$ and $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(V)$ with the (respective) continuous linear map:

$$\backslash begin\{alignat\}\{9\}$$

\,&& \mathcal{D}(U \times V) & \to \,&& \mathcal{D}(V) && \quad \text{ and } \quad && \,& \mathcal{D}(U \times V) && \to \,&& \mathcal{D}(U) \\

&& f \ & \mapsto\,&& \langle S, f^{\bullet} \rangle && && & f \ && \mapsto\,&& \langle T, f_{\bullet} \rangle \\

\end{alignat}

Moreover, if either $S$ (resp. $T$) has compact support then it also induces a continuous linear map of $C^\backslash infty(U\; \backslash times\; V)\; \backslash to\; C^\backslash infty(V)$ (resp. {{nowrap|$C^\backslash infty(U\; \backslash times\; V)\; \backslash to\; C^\backslash infty(U)$).}}{{sfn|Trèves|2006|pp=416-419}}

{{Math theorem|name={{visible anchor|Fubini's theorem for distributions|text=Fubini's theorem for distributions}}{{sfn|Trèves|2006|pp=416-419}}|math_statement=

Let $S\; \backslash in\; \backslash mathcal\{D\}\text{'}(U)$ and $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(V).$ If $f\; \backslash in\; \backslash mathcal\{D\}(U\; \backslash times\; V)$ then

$$\backslash langle\; S,\; \backslash langle\; T,\; f\_\{\backslash bullet\}\; \backslash rangle\; \backslash rangle\; =\; \backslash langle\; T,\; \backslash langle\; S,\; f^\{\backslash bullet\}\; \backslash rangle\; \backslash rangle.$$

}}

{{em|The Tensor product of $S\; \backslash in\; \backslash mathcal\{D\}\text{'}(U)$ and $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(V),$}} denoted by $S\; \backslash otimes\; T$ or $T\; \backslash otimes\; S,$ is the distribution in $U\; \backslash times\; V$ defined by:{{sfn|Trèves|2006|pp=416-419}}

$$(S\; \backslash otimes\; T)(f)\; :=\; \backslash langle\; S,\; \backslash langle\; T,\; f\_\{\backslash bullet\}\; \backslash rangle\; \backslash rangle\; =\; \backslash langle\; T,\; \backslash langle\; S,\; f^\{\backslash bullet\}\backslash rangle\; \backslash rangle.$$

{{See also|Spaces of test functions and distributions}}

For all and all every one of the following canonical injections is continuous and has an image (also called the range) that is a dense subset of its codomain:

$$\backslash begin\{matrix\}$$

C_c^\infty(U) & \to & C_c^k(U) & \to & C_c^0(U) & \to & L_c^\infty(U) & \to & L_c^p(U) & \to & L_c^1(U) \\

\downarrow & &\downarrow && \downarrow \\

C^\infty(U) & \to & C^k(U) & \to & C^0(U) \\{}

\end{matrix}

where the topologies on $L\_c^q(U)$ ($1\; \backslash leq\; q\; \backslash leq\; \backslash infty$) are defined as direct limits of the spaces $L\_c^q(K)$ in a manner analogous to how the topologies on $C\_c^k(U)$ were defined (so in particular, they are not the usual norm topologies). The range of each of the maps above (and of any composition of the maps above) is dense in its codomain.{{sfn|Trèves|2006|pp=150-160}}

Suppose that $X$ is one of the spaces $C\_c^k(U)$ (for $k\; \backslash in\; \backslash \{0,\; 1,\; \backslash ldots,\; \backslash infty\backslash \}$) or $L^p\_c(U)$ (for $1\; \backslash leq\; p\; \backslash leq\; \backslash infty$) or $L^p(U)$ (for ). Because the canonical injection $\backslash operatorname\{In\}\_X\; :\; C\_c^\backslash infty(U)\; \backslash to\; X$ is a continuous injection whose image is dense in the codomain, this map's transpose $\{\}^\{t\}\backslash operatorname\{In\}\_X\; :\; X\text{'}\_b\; \backslash to\; \backslash mathcal\{D\}\text{'}(U)\; =\; \backslash left(C\_c^\backslash infty(U)\backslash right)\text{'}\_b$ is a continuous injection. This injective transpose map thus allows the continuous dual space $X\text{'}$ of $X$ to be identified with a certain vector subspace of the space $\backslash mathcal\{D\}\text{'}(U)$ of all distributions (specifically, it is identified with the image of this transpose map). This transpose map is continuous but it is {{em|not}} necessarily a topological embedding.

A linear subspace of $\backslash mathcal\{D\}\text{'}(U)$ carrying a locally convex topology that is finer than the subspace topology induced on it by $\backslash mathcal\{D\}\text{'}(U)\; =\; \backslash left(C\_c^\backslash infty(U)\backslash right)\text{'}\_b$ is called {{em|a space of distributions}}.{{sfn|Trèves|2006|pp=240-252}}

Almost all of the spaces of distributions mentioned in this article arise in this way (for example, tempered distribution, restrictions, distributions of order $\backslash leq$ some integer, distributions induced by a positive Radon measure, distributions induced by an $L^p$-function, etc.) and any representation theorem about the continuous dual space of $X$ may, through the transpose $\{\}^\{t\}\backslash operatorname\{In\}\_X\; :\; X\text{'}\_b\; \backslash to\; \backslash mathcal\{D\}\text{'}(U),$ be transferred directly to elements of the space $\backslash operatorname\{Im\}\; \backslash left(\{\}^\{t\}\backslash operatorname\{In\}\_X\backslash right).$

The inclusion map $\backslash operatorname\{In\}\; :\; C\_c^\backslash infty(U)\; \backslash to\; C\_c^0(U)$ is a continuous injection whose image is dense in its codomain, so the transpose $\{\}^\{t\}\backslash operatorname\{In\}\; :\; (C\_c^0(U))\text{'}\_b\; \backslash to\; \backslash mathcal\{D\}\text{'}(U)\; =\; (C\_c^\backslash infty(U))\text{'}\_b$ is also a continuous injection.

Note that the continuous dual space $(C\_c^0(U))\text{'}\_b$ can be identified as the space of Radon measures, where there is a one-to-one correspondence between the continuous linear functionals $T\; \backslash in\; (C\_c^0(U))\text{'}\_b$ and integral with respect to a Radon measure; that is,

• if $T\; \backslash in\; (C\_c^0(U))\text{'}\_b$ then there exists a Radon measure $\backslash mu$ on {{mvar|U}} such that for all $f\; \backslash in\; C\_c^0(U),\; T(f)\; =\; \backslash int\_U\; f\; \backslash ,\; d\backslash mu,$ and
• if $\backslash mu$ is a Radon measure on {{mvar|U}} then the linear functional on $C\_c^0(U)$ defined by sending $f\; \backslash in\; C\_c^0(U)$ to $\backslash int\_U\; f\; \backslash ,\; d\backslash mu$ is continuous.

Through the injection $\{\}^\{t\}\backslash operatorname\{In\}\; :\; (C\_c^0(U))\text{'}\_b\; \backslash to\; \backslash mathcal\{D\}\text{'}(U),$ every Radon measure becomes a distribution on {{mvar|U}}. If $f$ is a locally integrable function on {{mvar|U}} then the distribution $\backslash phi\; \backslash mapsto\; \backslash int\_U\; f(x)\; \backslash phi(x)\; \backslash ,\; dx$ is a Radon measure; so Radon measures form a large and important space of distributions.

The following is the theorem of the structure of distributions of Radon measures, which shows that every Radon measure can be written as a sum of derivatives of locally $L^\backslash infty$ functions on {{mvar|U}}:

{{math theorem|name=Theorem.{{sfn|Trèves|2006|pp=262–264}}|math_statement=

Suppose $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(U)$ is a Radon measure, where $U\; \backslash subseteq\; \backslash R^n,$ let $V\; \backslash subseteq\; U$ be a neighborhood of the support of $T,$ and let $I\; =\; \backslash \{p\; \backslash in\; \backslash N^n\; :\; |p|\; \backslash leq\; n\backslash \}.$ There exists a family $f=(f\_p)\_\{p\backslash in\; I\}$ of locally $L^\backslash infty$ functions on {{mvar|U}} such that $\backslash operatorname\{supp\}\; f\_p\; \backslash subseteq\; V$ for every $p\backslash in\; I,$ and

$$T\; =\; \backslash sum\_\{p\backslash in\; I\}\; \backslash partial^p\; f\_p.$$

Furthermore, $T$ is also equal to a finite sum of derivatives of continuous functions on $U,$ where each derivative has order $\backslash leq\; 2\; n.$

}}

A linear function $T$ on a space of functions is called {{em|positive}} if whenever a function $f$ that belongs to the domain of $T$ is non-negative (that is, $f$ is real-valued and $f\; \backslash geq\; 0$) then $T(f)\; \backslash geq\; 0.$ One may show that every positive linear functional on $C\_c^0(U)$ is necessarily continuous (that is, necessarily a Radon measure).{{sfn|Trèves|2006|p=218}}

Lebesgue measure is an example of a positive Radon measure.

One particularly important class of Radon measures are those that are induced locally integrable functions. The function $f\; :\; U\; \backslash to\; \backslash R$ is called {{em|locally integrable}} if it is Lebesgue integrable over every compact subset {{mvar|K}} of {{mvar|U}}. This is a large class of functions that includes all continuous functions and all Lp space $L^p$ functions. The topology on $\backslash mathcal\{D\}(U)$ is defined in such a fashion that any locally integrable function $f$ yields a continuous linear functional on $\backslash mathcal\{D\}(U)$ – that is, an element of $\backslash mathcal\{D\}\text{'}(U)$ – denoted here by $T\_f,$ whose value on the test function $\backslash phi$ is given by the Lebesgue integral:

$$\backslash langle\; T\_f,\; \backslash phi\; \backslash rangle\; =\; \backslash int\_U\; f\; \backslash phi\backslash ,dx.$$

Conventionally, one abuses notation by identifying $T\_f$ with $f,$ provided no confusion can arise, and thus the pairing between $T\_f$ and $\backslash phi$ is often written

$$\backslash langle\; f,\; \backslash phi\; \backslash rangle\; =\; \backslash langle\; T\_f,\; \backslash phi\; \backslash rangle.$$

If $f$ and $g$ are two locally integrable functions, then the associated distributions $T\_f$ and $T\_g$ are equal to the same element of $\backslash mathcal\{D\}\text{'}(U)$ if and only if $f$ and $g$ are equal almost everywhere (see, for instance, {{harvtxt|Hörmander|1983|loc=Theorem 1.2.5}}). Similarly, every Radon measure $\backslash mu$ on $U$ defines an element of $\backslash mathcal\{D\}\text{'}(U)$ whose value on the test function $\backslash phi$ is $\backslash int\backslash phi\; \backslash ,d\backslash mu.$ As above, it is conventional to abuse notation and write the pairing between a Radon measure $\backslash mu$ and a test function $\backslash phi$ as $\backslash langle\; \backslash mu,\; \backslash phi\; \backslash rangle.$ Conversely, as shown in a theorem by Schwartz (similar to the Riesz representation theorem), every distribution which is non-negative on non-negative functions is of this form for some (positive) Radon measure.

The test functions are themselves locally integrable, and so define distributions. The space of test functions $C\_c^\backslash infty(U)$ is sequentially dense in $\backslash mathcal\{D\}\text{'}(U)$ with respect to the strong topology on $\backslash mathcal\{D\}\text{'}(U).${{sfn|Trèves|2006|pp=300-304}} This means that for any $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(U),$ there is a sequence of test functions, $(\backslash phi\_i)\_\{i=1\}^\backslash infty,$ that converges to $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(U)$ (in its strong dual topology) when considered as a sequence of distributions. Or equivalently,

$$\backslash langle\; \backslash phi\_i,\; \backslash psi\; \backslash rangle\; \backslash to\; \backslash langle\; T,\; \backslash psi\; \backslash rangle\; \backslash qquad\; \backslash text\{\; for\; all\; \}\; \backslash psi\; \backslash in\; \backslash mathcal\{D\}(U).$$

The inclusion map $\backslash operatorname\{In\}:\; C\_c^\backslash infty(U)\; \backslash to\; C^\backslash infty(U)$ is a continuous injection whose image is dense in its codomain, so the transpose map $\{\}^\{t\}\backslash operatorname\{In\}:\; (C^\backslash infty(U))\text{'}\_b\; \backslash to\; \backslash mathcal\{D\}\text{'}(U)\; =\; (C\_c^\backslash infty(U))\text{'}\_b$ is also a continuous injection. Thus the image of the transpose, denoted by $\backslash mathcal\{E\}\text{'}(U),$ forms a space of distributions.{{sfn|Trèves|2006|pp=255-257}}

The elements of $\backslash mathcal\{E\}\text{'}(U)\; =\; (C^\backslash infty(U))\text{'}\_b$ can be identified as the space of distributions with compact support.{{sfn|Trèves|2006|pp=255-257}} Explicitly, if $T$ is a distribution on {{mvar|U}} then the following are equivalent,

• $T\; \backslash in\; \backslash mathcal\{E\}\text{'}(U).$
• The support of $T$ is compact.
• The restriction of $T$ to $C\_c^\backslash infty(U),$ when that space is equipped with the subspace topology inherited from $C^\backslash infty(U)$ (a coarser topology than the canonical LF topology), is continuous.{{sfn|Trèves|2006|pp=255-257}}
• There is a compact subset {{mvar|K}} of {{mvar|U}} such that for every test function $\backslash phi$ whose support is completely outside of {{mvar|K}}, we have $T(\backslash phi)\; =\; 0.$

Compactly supported distributions define continuous linear functionals on the space $C^\backslash infty(U)$; recall that the topology on $C^\backslash infty(U)$ is defined such that a sequence of test functions $\backslash phi\_k$ converges to 0 if and only if all derivatives of $\backslash phi\_k$ converge uniformly to 0 on every compact subset of {{mvar|U}}. Conversely, it can be shown that every continuous linear functional on this space defines a distribution of compact support. Thus compactly supported distributions can be identified with those distributions that can be extended from $C\_c^\backslash infty(U)$ to $C^\backslash infty(U).$

Let $k\; \backslash in\; \backslash N.$ The inclusion map $\backslash operatorname\{In\}:\; C\_c^\backslash infty(U)\; \backslash to\; C\_c^k(U)$ is a continuous injection whose image is dense in its codomain, so the transpose $\{\}^\{t\}\backslash operatorname\{In\}:\; (C\_c^k(U))\text{'}\_b\; \backslash to\; \backslash mathcal\{D\}\text{'}(U)\; =\; (C\_c^\backslash infty(U))\text{'}\_b$ is also a continuous injection. Consequently, the image of $\{\}^\{t\}\backslash operatorname\{In\},$ denoted by $\backslash mathcal\{D\}\text{'}^\{k\}(U),$ forms a space of distributions. The elements of $\backslash mathcal\{D\}\text{'}^k(U)$ are {{em|the distributions of order $\backslash ,\backslash leq\; k.$}}{{sfn|Trèves|2006|pp=258-264}} The distributions of order $\backslash ,\backslash leq\; 0,$ which are also called {{em|distributions of order {{math|0}}}} are exactly the distributions that are Radon measures (described above).

For $0\; \backslash neq\; k\; \backslash in\; \backslash N,$ a {{em|distribution of order {{mvar|k}}}} is a distribution of order $\backslash ,\backslash leq\; k$ that is not a distribution of order $\backslash ,\backslash leq\; k\; -\; 1$.{{sfn|Trèves|2006|pp=258-264}}

A distribution is said to be of {{em|finite order}} if there is some integer $k$ such that it is a distribution of order $\backslash ,\backslash leq\; k,$ and the set of distributions of finite order is denoted by $\backslash mathcal\{D\}\text{'}^\{F\}(U).$ Note that if $k\; \backslash leq\; l$ then $\backslash mathcal\{D\}\text{'}^k(U)\; \backslash subseteq\; \backslash mathcal\{D\}\text{'}^l(U)$ so that $\backslash mathcal\{D\}\text{'}^\{F\}(U)\; :=\; \backslash bigcup\_\{n=0\}^\backslash infty\; \backslash mathcal\{D\}\text{'}^n(U)$ is a vector subspace of $\backslash mathcal\{D\}\text{'}(U)$, and furthermore, if and only if $\backslash mathcal\{D\}\text{'}^\{F\}(U)\; =\; \backslash mathcal\{D\}\text{'}(U).${{sfn|Trèves|2006|pp=258-264}}

Every distribution with compact support in {{mvar|U}} is a distribution of finite order.{{sfn|Trèves|2006|pp=258-264}} Indeed, every distribution in {{mvar|U}} is {{em|locally}} a distribution of finite order, in the following sense:{{sfn|Trèves|2006|pp=258-264}} If {{mvar|V}} is an open and relatively compact subset of {{mvar|U}} and if $\backslash rho\_\{VU\}$ is the restriction mapping from {{mvar|U}} to {{mvar|V}}, then the image of $\backslash mathcal\{D\}\text{'}(U)$ under $\backslash rho\_\{VU\}$ is contained in $\backslash mathcal\{D\}\text{'}^\{F\}(V).$

The following is the theorem of the structure of distributions of finite order, which shows that every distribution of finite order can be written as a sum of derivatives of Radon measures:

{{math theorem|name=Theorem{{sfn|Trèves|2006|pp=258-264}}|math_statement=Suppose $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(U)$ has finite order and $I\; =\backslash \{p\; \backslash in\; \backslash N^n\; :\; |p|\; \backslash leq\; k\backslash \}.$ Given any open subset {{mvar|V}} of {{mvar|U}} containing the support of $T,$ there is a family of Radon measures in {{mvar|U}}, $(\backslash mu\_p)\_\{p\; \backslash in\; I\},$ such that for very $p\; \backslash in\; I,\; \backslash operatorname\{supp\}(\backslash mu\_p)\; \backslash subseteq\; V$ and

$$T\; =\; \backslash sum\_\{|p|\; \backslash leq\; k\}\; \backslash partial^p\; \backslash mu\_p.$$}}

Example. (Distributions of infinite order) Let $U\; :=\; (0,\; \backslash infty)$ and for every test function $f,$ let $$S\; f\; :=\; \backslash sum\_\{m=1\}^\backslash infty\; (\backslash partial^m\; f)\backslash left(\backslash frac\{1\}\{m\}\backslash right).$$

Then $S$ is a distribution of infinite order on {{mvar|U}}. Moreover, $S$ can not be extended to a distribution on $\backslash R$; that is, there exists no distribution $T$ on $\backslash R$ such that the restriction of $T$ to {{mvar|U}} is equal to $S.${{sfn|Rudin|1991|pp=177-181}}

{{Redirect|Tempered distribution|tempered distributions on semisimple groups|Tempered representation}}

Defined below are the {{em|tempered distributions}}, which form a subspace of $\backslash mathcal\{D\}\text{'}(\backslash R^n),$ the space of distributions on $\backslash R^n.$ This is a proper subspace: while every tempered distribution is a distribution and an element of $\backslash mathcal\{D\}\text{'}(\backslash R^n),$ the converse is not true. Tempered distributions are useful if one studies the Fourier transform since all tempered distributions have a Fourier transform, which is not true for an arbitrary distribution in $\backslash mathcal\{D\}\text{'}(\backslash R^n).$

The Schwartz space $\backslash mathcal\{S\}(\backslash R^n)$ is the space of all smooth functions that are rapidly decreasing at infinity along with all partial derivatives. Thus $\backslash phi:\backslash R^n\backslash to\backslash R$ is in the Schwartz space provided that any derivative of $\backslash phi,$ multiplied with any power of $|x|,$ converges to 0 as $|x|\; \backslash to\; \backslash infty.$ These functions form a complete TVS with a suitably defined family of seminorms. More precisely, for any multi-indices $\backslash alpha$ and $\backslash beta$ define

$$p\_\{\backslash alpha,\; \backslash beta\}(\backslash phi)\; =\; \backslash sup\_\{x\; \backslash in\; \backslash R^n\}\; \backslash left|x^\backslash alpha\; \backslash partial^\backslash beta\; \backslash phi(x)\; \backslash right|.$$

Then $\backslash phi$ is in the Schwartz space if all the values satisfy

The family of seminorms $p\_\{\backslash alpha,\backslash beta\}$ defines a locally convex topology on the Schwartz space. For $n\; =\; 1,$ the seminorms are, in fact, norms on the Schwartz space. One can also use the following family of seminorms to define the topology:{{sfn|Trèves|2006|pp=92-94}}

$$|f|\_\{m,k\}\; =\; \backslash sup\_\{|p|\backslash le\; m\}\; \backslash left(\backslash sup\_\{x\; \backslash in\; \backslash R^n\}\; \backslash left\backslash \{(1\; +\; |x|)^k\; \backslash left|(\backslash partial^\backslash alpha\; f)(x)\; \backslash right|\backslash right\backslash \}\backslash right),\; \backslash qquad\; k,m\; \backslash in\; \backslash N.$$

Otherwise, one can define a norm on $\backslash mathcal\{S\}(\backslash R^n)$ via

$$\backslash |\backslash phi\backslash |\_k\; =\; \backslash max\_\{|\backslash alpha|\; +\; |\backslash beta|\; \backslash leq\; k\}\; \backslash sup\_\{x\; \backslash in\; \backslash R^n\}\; \backslash left|\; x^\backslash alpha\; \backslash partial^\backslash beta\; \backslash phi(x)\backslash right|,\; \backslash qquad\; k\; \backslash ge\; 1.$$

The Schwartz space is a Fréchet space (that is, a complete metrizable locally convex space). Because the Fourier transform changes $\backslash partial^\backslash alpha$ into multiplication by $x^\backslash alpha$ and vice versa, this symmetry implies that the Fourier transform of a Schwartz function is also a Schwartz function.

A sequence $\backslash \{f\_i\backslash \}$ in $\backslash mathcal\{S\}(\backslash R^n)$ converges to 0 in $\backslash mathcal\{S\}(\backslash R^n)$ if and only if the functions $(1\; +\; |x|)^k\; (\backslash partial^p\; f\_i)(x)$ converge to 0 uniformly in the whole of $\backslash R^n,$ which implies that such a sequence must converge to zero in $C^\backslash infty(\backslash R^n).${{sfn|Trèves|2006|pp=92–94}}

$\backslash mathcal\{D\}(\backslash R^n)$ is dense in $\backslash mathcal\{S\}(\backslash R^n).$ The subset of all analytic Schwartz functions is dense in $\backslash mathcal\{S\}(\backslash R^n)$ as well.{{sfn|Trèves|2006|p=160}}

The Schwartz space is nuclear, and the tensor product of two maps induces a canonical surjective TVS-isomorphisms

$$\backslash mathcal\{S\}(\backslash R^m)\backslash \; \backslash widehat\{\backslash otimes\}\backslash \; \backslash mathcal\{S\}(\backslash R^n)\; \backslash to\; \backslash mathcal\{S\}(\backslash R^\{m+n\}),$$

where $\backslash widehat\{\backslash otimes\}$ represents the completion of the injective tensor product (which in this case is identical to the completion of the projective tensor product).{{sfn|Trèves|2006|p=531}}

The inclusion map $\backslash operatorname\{In\}:\; \backslash mathcal\{D\}(\backslash R^n)\; \backslash to\; \backslash mathcal\{S\}(\backslash R^n)$ is a continuous injection whose image is dense in its codomain, so the transpose $\{\}^\{t\}\backslash operatorname\{In\}:\; (\backslash mathcal\{S\}(\backslash R^n))\text{'}\_b\; \backslash to\; \backslash mathcal\{D\}\text{'}(\backslash R^n)$ is also a continuous injection. Thus, the image of the transpose map, denoted by $\backslash mathcal\{S\}\text{'}(\backslash R^n),$ forms a space of distributions.

The space $\backslash mathcal\{S\}\text{'}(\backslash R^n)$ is called the space of {{em|tempered distributions}}. It is the continuous dual space of the Schwartz space. Equivalently, a distribution $T$ is a tempered distribution if and only if

$$\backslash left(\backslash text\{\; for\; all\; \}\; \backslash alpha,\; \backslash beta\; \backslash in\; \backslash N^n:\; \backslash lim\_\{m\backslash to\; \backslash infty\}\; p\_\{\backslash alpha,\; \backslash beta\}\; (\backslash phi\_m)\; =\; 0\; \backslash right)\; \backslash Longrightarrow\; \backslash lim\_\{m\backslash to\; \backslash infty\}\; T(\backslash phi\_m)=0.$$

The derivative of a tempered distribution is again a tempered distribution. Tempered distributions generalize the bounded (or slow-growing) locally integrable functions; all distributions with compact support and all square-integrable functions are tempered distributions. More generally, all functions that are products of polynomials with elements of Lp space $L^p(\backslash R^n)$ for $p\; \backslash geq\; 1$ are tempered distributions.

The {{em|tempered distributions}} can also be characterized as {{em|slowly growing}}, meaning that each derivative of $T$ grows at most as fast as some polynomial. This characterization is dual to the {{em|rapidly falling}} behaviour of the derivatives of a function in the Schwartz space, where each derivative of $\backslash phi$ decays faster than every inverse power of $|x|.$ An example of a rapidly falling function is $|x|^n\backslash exp\; (-\backslash lambda\; |x|^\backslash beta)$ for any positive $n,\; \backslash lambda,\; \backslash beta.$

To study the Fourier transform, it is best to consider complex-valued test functions and complex-linear distributions. The ordinary continuous Fourier transform $F\; :\; \backslash mathcal\{S\}(\backslash R^n)\; \backslash to\; \backslash mathcal\{S\}(\backslash R^n)$ is a TVS-automorphism of the Schwartz space, and the {{em|Fourier transform}} is defined to be its transpose $\{\}^\{t\}F\; :\; \backslash mathcal\{S\}\text{'}(\backslash R^n)\; \backslash to\; \backslash mathcal\{S\}\text{'}(\backslash R^n),$ which (abusing notation) will again be denoted by $F.$ So the Fourier transform of the tempered distribution $T$ is defined by $(FT)(\backslash psi)\; =\; T(F\; \backslash psi)$ for every Schwartz function $\backslash psi.$ $FT$ is thus again a tempered distribution. The Fourier transform is a TVS isomorphism from the space of tempered distributions onto itself. This operation is compatible with differentiation in the sense that

$$F\; \backslash dfrac\{dT\}\{dx\}\; =\; ixFT$$

and also with convolution: if $T$ is a tempered distribution and $\backslash psi$ is a {{em|slowly increasing}} smooth function on $\backslash R^n,$ $\backslash psi\; T$ is again a tempered distribution and

$$F(\backslash psi\; T)\; =\; F\; \backslash psi\; *\; FT$$

is the convolution of $FT$ and $F\; \backslash psi.$ In particular, the Fourier transform of the constant function equal to 1 is the $\backslash delta$ distribution.

If $T\; \backslash in\; \backslash mathcal\{S\}\text{'}(\backslash R^n)$ is a tempered distribution, then there exists a constant $C\; >\; 0,$ and positive integers $M$ and $N$ such that for all Schwartz functions $\backslash phi\; \backslash in\; \backslash mathcal\{S\}(\backslash R^n)$

$$\backslash langle\; T,\; \backslash phi\; \backslash rangle\; \backslash le\; C\backslash sum\backslash nolimits\_\{|\backslash alpha|\backslash le\; N,\; |\backslash beta|\backslash le\; M\}\backslash sup\_\{x\; \backslash in\; \backslash R^n\}\; \backslash left|x^\backslash alpha\; \backslash partial^\backslash beta\; \backslash phi(x)\; \backslash right|=C\backslash sum\backslash nolimits\_\{|\backslash alpha|\backslash le\; N,\; |\backslash beta|\backslash le\; M\}\; p\_\{\backslash alpha,\; \backslash beta\}(\backslash phi).$$

This estimate, along with some techniques from functional analysis, can be used to show that there is a continuous slowly increasing function $F$ and a multi-index $\backslash alpha$ such that

$$T\; =\; \backslash partial^\backslash alpha\; F.$$

If $T\; \backslash in\; \backslash mathcal\{D\}\text{'}(\backslash R^n),$ then for any compact set $K\; \backslash subseteq\; \backslash R^n,$ there exists a continuous function $F$compactly supported in $\backslash R^n$ (possibly on a larger set than {{mvar|K}} itself) and a multi-index $\backslash alpha$ such that $T\; =\; \backslash partial^\backslash alpha\; F$ on $C\_c^\backslash infty(K).$

The success of the theory led to an investigation of the idea of hyperfunction, in which spaces of holomorphic functions are used as test functions. A refined theory has been developed, in particular Mikio Sato's algebraic analysis, using sheaf theory and several complex variables. This extends the range of symbolic methods that can be made into rigorous mathematics, for example, Feynman integrals.

• {{annotated link|Cauchy principal value}}
• {{annotated link|Gelfand triple}}
• {{annotated link|Gelfand–Shilov space}}
• {{annotated link|Generalized function}}
• {{annotated link|Hilbert transform}}
• {{annotated link|Homogeneous distribution}}
• {{annotated link|Laplacian of the indicator}}
• {{annotated link|Limit of distributions}}
• {{annotated link|Mollifier}}
• {{annotated link|Vague topology}}
• {{annotated link|Ultradistribution}}

Differential equations related

• {{annotated link|Fundamental solution}}
• {{annotated link|Pseudo-differential operator}}
• {{annotated link|Weak solution}}

Generalizations of distributions

• {{annotated link|Colombeau algebra}}
• {{annotated link|Current (mathematics)}}
• {{annotated link|Distribution (number theory)}}
• {{annotated link|Distribution on a linear algebraic group}}
• {{annotated link|Green's function}}
• {{annotated link|Hyperfunction}}
• {{annotated link|Malgrange–Ehrenpreis theorem}}

{{reflist|group=note}}

{{reflist|29em}}

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• M. J. Lighthill (1959). Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press. {{ISBN|0-521-09128-4}} (requires very little knowledge of analysis; defines distributions as limits of sequences of functions under integrals)
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• {{springer|id=G/g130030|title=Generalized function algebras|first=Michael|last=Oberguggenberger|year=2001}}.

{{Functional analysis}}

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