Distributions, also known as Schwartz distributions or generalized functions
, are objects that generalize the classical notion of functions in mathematical analysis
. Distributions make it possible to differentiate
functions whose derivatives do not exist in the classical sense. In particular, any locally integrable
function has a distributional derivative. Distributions are widely used in the theory of partial differential equation
s, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist. Distributions are also important in physics
where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta
is normally thought of as ''acting''
on the ''points'' in its domain
by "sending" a point in its domain to the point
Instead of acting on points, distribution theory reinterprets functions such as
as acting on ''test functions'' in a certain way. ''Test functions'' are usually infinitely differentiable complex
-valued (or sometimes real
-valued) functions with compact support
s are examples of test functions). Many "standard functions" (meaning for example a function that is typically encountered in a Calculus
course), say for instance a continuous
can be canonically reinterpreted as acting on test functions (instead of their usual interpretation as acting on points of their domain) via the action known as "integration
against a test function"; explicitly, this means that
"acts on" a test function by "sending" to the number
This new action of
is thus a complex (or real)-valued map
, denoted by
whose domain is the space
of test functions; this map turns out to have two additional properties
[ turns out to also be linear and continuous when the space of test functions is given a certain topology called ''the canonical LF topology''.]
that make it into what is known as a ''distribution on
'' Distributions that arise from "standard functions" in this way are the prototypical examples of a distributions. But there are many distributions that do not arise in this way and these distributions are known as "generalized functions." Examples include the Dirac delta function
or some distributions that arise via the action of "integration of test functions against measures
." However, by using various methods it is nevertheless still possible to reduce any arbitrary distribution
down to a simpler ''family'' of related distributions that do arise via such actions of integration.
In applications to physics and engineering, the space of test functions usually consists of smooth functions with compact support
that are defined on some given non-empty open subset
This space of test functions is denoted by
and a ''distribution on '' is by definition a linear functional
that is continuous
is given a topology called ''the canonical LF topology''. This leads to ''the'' space of (all) distributions on , usually denoted by
(note the prime
), which by definition is the space
of all distributions on
(that is, it is the continuous dual space
); it is these distributions that are the main focus of this article.
There are other possible choices for the space of test functions, which lead to other different space
s of distributions. If
then the use of Schwartz functions
[The Schwartz space consists of smooth rapidly decreasing test functions, where "rapidly decreasing" means that the function decreases faster than any polynomial increases as points in its domain move away from the origin.]
as test functions gives rise to a certain subspace of
whose elements are called ''tempered distributions''. These are important because they allow the Fourier transform
to be extended from "standard functions" to tempered distributions. The set of tempered distributions forms a vector subspace
of space of distributions
and is thus one example of space of distributions; there are many other spaces of distributions.
There also exist other major classes of test functions that are ''not'' subsets of
such as spaces of analytic test functions
, which produce very different classes of distributions. The theory of such distributions has a different character from the previous one because there are no analytic functions with non-empty compact support.
[Except for the trivial (i.e. identically ) map, which of course is always analytic.]
Use of analytic test functions lead to Sato
's theory of hyperfunction
The practical use of distributions can be traced back to the use of Green functions
in the 1830s to solve ordinary differential equations, but was not formalized until much later. According to , generalized functions originated in the work of 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. comments that although the ideas in the transformative book by 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.
The following notation will be used throughout this article:
is a fixed positive integer and
is a fixed non-empty open subset
of Euclidean space
denotes the natural number
will denote a non-negative integer or
is a function
will denote its domain
and the ''support
'' denoted by
is defined to be the closure
of the set
* For two functions
, the following notation defines a canonical pairing
* A ''multi-index
'' of size
is an element in
is fixed, if the size of multi-indices is omitted then the size should be assumed to be
). The ''length'' of a multi-index
is defined as
and denoted by
Multi-indices are particularly useful when dealing with functions of several variables, in particular we introduce the following notations for a given multi-index
:We also introduce a partial order of all multi-indices by
if and only if
we define their multi-index binomial coefficient as:
will denote a certain non-empty collection of compact subsets of
(described in detail below).
Definitions of test functions and distributions
In this section, we will formally define real-valued distributions on . With minor modifications, one can also define complex-valued distributions, and one can replace
with any (paracompact
) smooth manifold
Note that for all
and any compact subsets and of , we have:
Distributions on are defined to be the continuous linear functional
when this vector space is endowed with a particular topology called the ''canonical LF-topology''.
This topology is unfortunately not easy to define but it is nevertheless still possible to characterize distributions in a way so that no mention of the canonical LF-topology is made.
Proposition: If is a linear functional
then the is a distribution if and only if the following equivalent conditions are satisfied:
# For every compact subset
there exist constants
such that for all
# For every compact subset
there exist constants
such that for all
with support contained in
# For any compact subset
and any sequence
converges uniformly to zero on
for all multi-indices
The above characterizations can be used to determine whether or not a linear functional is a distribution, but more advanced uses of distributions and test functions (such as applications to differential equations
) is limited if no topologies are placed on
To define the space of distributions we must first define the canonical LF-topology, which in turn requires that several other topological vector space
s (TVSs) be defined first. We will first define a topology on
then assign every
the subspace topology
induced on it by
and finally we define the canonical LF-topology on
We use the canonical LF-topology to define a topology on the space of distributions, which permits us to consider things such as convergence of distributions.
;Choice of compact sets
Throughout, will be any collection of compact subsets of such that (1)
and (2) for any compact there exists some such that . The most common choices for are:
* The set of all compact subsets of , or
* A set
and for all ,
and is a relatively compact
non-empty open subset of (i.e. "relatively compact" means that the closure
of , in either or
We make into a directed set
by defining if and only if . Note that although the definitions of the subsequently defined topologies explicitly reference , in reality they do not depend on the choice of ; that is, if and are any two such collections of compact subsets of , then the topologies defined on
by using in place of are the same as those defined by using in place of .
Topology on Ck(U)
We now introduce the seminorm
s that will define the topology on
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.
Each of the functions above are non-negative -valued
[The image of the compact set under a continuous -valued map (e.g. for ) is itself a compact, and thus bounded, subset of . If then this implies that each of the functions defined above is -valued (i.e. none of the supremums above are ever equal to ).] seminorm
Each of the following families of seminorms generates the same locally convex vector topology
With this topology,
becomes a locally convex (''non''-normable
) Fréchet space
and all of the seminorms defined above are continuous on this space. ''All'' of the seminorms defined above are continuous functions on
Under this topology, a net
if and only if for every multi-index with and every , the net
uniformly on . For any
any bounded subset
is a relatively compact
In particular, a subset of
is bounded if and only if it is bounded in
is a Montel space
if and only if .
The topology on
is the superior limit of the subspace topologies
by the TVSs
as ranges over the non-negative integers. A subset of
is open in this topology if and only if there exists
such that is open when
is endowed with the subspace topology
;Metric defining the topology
If the family of compact sets
for all , then a complete translation-invariant metric on
can be obtained by taking a suitable countable Fréchet combination
of any one of the above families.
For example, using the seminorms
Often, it is easier to just consider seminorms.
Topology on Ck(K)
As before, fix
Recall that if
is any compact subset of
For any compact subset ,
is a closed subspace of the Fréchet space
and is thus also a Fréchet space
. For all compact with , denote the natural inclusion by
Then this map is a linear embedding of TVSs (i.e. a linear map that is also a topological embedding
) whose range is closed in its codomain
; said differently, the topology on
is identical to the subspace topology it inherits from
is a closed subset of
is finite then
is a Banach space
with a topology that can be defined by the norm
And when , then
is even a Hilbert space
. The space
is a distinguished Schwartz Montel space
then it is ''not'' normable
and thus ''not'' a Banach space (although like all other
it is a Fréchet space
Trivial extensions and independence of ''C''''k''(''K'')'s topology from ''U''
The definition of
depends on so we will let
denote the topological space
which by definition is a topological subspace
Suppose is an open subset of
its is by definition, the function
denote the map that sends a function in
to its trivial extension on . This map is a linear injection
and for every compact subset
is the vector subspace of
consisting of maps with support contained in (since , is a compact subset of as well). It follows that
If is restricted to
then the following induced linear map is a homeomorphism
(and thus a TVS-isomorphism):
and thus the next two maps (which like the previous map are defined by
) are topological embedding
(the topology on
is the canonical LF topology, which is defined later). Using
with its image in
through this identification,
can also be considered as a subset of
Importantly, the subspace topology
(when it is viewed as a subset of
) is identical to the subspace topology that it inherits from
is viewed instead as a subset of
via the identification). Thus the topology on
is independent of the open subset of
that contains . This justifies the practice of written
Topology on the spaces of test functions and distributions
denote all those functions in
that have compact support in , where note that
is the union of all
as ranges over . Moreover, for every ,
is a dense subset of
The special case when gives us the space of test functions.
Canonical LF topology
We now define the canonical LF topology as a direct limit
. It is also possible to define this topology in terms of its neighborhoods of the origin, which is described afterwards.
For any two sets and , we declare that if and only if , which in particular makes the collection of compact subsets of into a directed set
(we say that such a collection is ''directed by subset inclusion''). For all compact with , there are natural inclusion
Recall from above that the map
is a topological embedding
. The collection of maps
forms a direct system
in the category
of locally convex topological vector space
s that is directed
by (under subset inclusion). This system's direct limit
(in the category of locally convex TVSs) is the pair
are the natural inclusions and where
is now endowed with the (unique) strongest
locally convex topology making all of the inclusion maps
;Neighborhoods of the origin
If is a convex
then is a neighborhood
of the origin in the canonical LF topology if and only if it satisfies the following condition:
Note that any convex set satisfying this condition is necessarily absorbing
Since the topology of any topological vector space
is translation-invariant, any TVS-topology is completely determined by the set of neighborhood of the origin. This means that one could actually ''define'' the canonical LF topology by declaring that a convex balanced
subset is a neighborhood of the origin if and only if it satisfies condition .
;Topology defined via differential operators
A ''linear differential operator in with smooth coefficients'' is a sum
and all but finitely many of
are identically . The integer
is called the ''order'' of the differential operator
is a linear differential operator of order then it induces a canonical linear map
where we shall reuse notation and also denote this map by
For any , the canonical LF topology on
is the weakest locally convex TVS topology making all linear differential operators in of order into continuous maps from
= Basic properties
;Canonical LF topology's independence from
One benefit of defining the canonical LF topology as the direct limit of a direct system
is that we may immediately use the universal property of direct limits. Another benefit is that we can use well-known results from category theory
to deduce that the canonical LF topology is actually independent of the particular choice of the directed
collection of compact sets. And by considering different collections (in particular, those mentioned at the beginning of this article), we may deduce different properties of this topology. In particular, we may deduce that the canonical LF topology makes
into a Hausdorff locally convex strict LF-space
(and also a strict LB-space
if ), which of course is the reason why this topology is called "the canonical LF topology" (see this footnote for more details).
[If we take to be the set of ''all'' compact subsets of then we can use the universal property of direct limits to conclude that the inclusion is a continuous and even that they are topological embedding for every compact subset . If however, we take to be the set of closures of some countable increasing sequence of relatively compact open subsets of having all of the properties mentioned earlier in this in this article then we immediately deduce that is a Hausdorff locally convex strict LF-space (and even a strict LB-space when ). All of these facts can also be proved directly without using direct systems (although with more work).]
From the universal property of direct limit
s, we know that if
is a linear map into a locally convex space (not necessarily Hausdorff), then is continuous if and only if is bounded
if and only if for every , the restriction of to
is continuous (or bounded).
;Dependence of the canonical LF topology on
Suppose is an open subset of
denote the map that sends a function in
to its trivial extension on (which was defined above). This map is a continuous linear map. If (and only if) then
is ''not'' a dense subset of
is ''not'' a topological embedding
. Consequently, if then the transpose of
is neither one-to-one nor onto.
A subset of
if and only if there exists some such that
and is a bounded subset of
Moreover, if is compact and
then is bounded in
if and only if it is bounded in
For any , any bounded subset of
) is a relatively compact
), where .
For all compact , the interior of
is empty so that
is of the first category in itself. It follows from Baire's theorem
is ''not'' metrizable
and thus also ''not'' normable
(see this footnote
[For any TVS (metrizable or otherwise), the notion of completeness depends entirely on a certain so-called "canonical uniformity" that is defined using ''only'' the subtraction operation (see the article Complete topological vector space for more details). In this way, the notion of a complete TVS does not ''require'' the existence of any metric. However, if the TVS is metrizable and if is ''any'' translation-invariant metric on that defines its topology, then is complete as a TVS (i.e. it is a complete uniform space under its canonical uniformity) if and only if the is a complete metric space. So if a TVS happens to have a topology that can be defined by such a metric then may be used to deduce the completeness of but the existence of such a metric is not necessary for defining completeness and it is even possible to deduce that a metrizable TVS is complete without ever even considering a metric (e.g. since the Cartesian product of any collection of complete TVSs is again a complete TVS, we can immediately deduce that the TVS which happens to be metrizable, is a complete TVS; note that there was no need to consider any metric on ).]
for an explanation of how the non-metrizable space
can be complete even thought it does not admit a metric). The fact that
is a nuclear Montel space
makes up for the non-metrizability of
(see this footnote for a more detailed explanation).
[One reason for giving the canonical LF topology is because it is with this topology that and its continuous dual space both become nuclear spaces, which have many nice properties and which may be viewed as a generalization of finite-dimensional spaces (for comparison, normed spaces are another generalization of finite-dimensional spaces that have many "nice" properties). In more detail, there are two classes of topological vector spaces (TVSs) that are particularly similar to finite-dimensional Euclidean spaces: the Banach spaces (especially Hilbert spaces) and the nuclear Montel spaces. Montel spaces are a class of TVSs in which every closed and bounded subset is compact (this generalizes the Heine–Borel theorem), which is a property that no infinite-dimensional Banach space can have; that is, no infinite-dimensional TVS can be both a Banach space and a Montel space. Also, no infinite-dimensional TVS can be both a Banach space and a nuclear space. All finite dimensional Euclidean spaces are nuclear Montel Hilbert spaces but once one enters infinite-dimensional space then these two classes separate. Nuclear spaces in particular have many of the "nice" properties of finite-dimensional TVSs (e.g. the Schwartz kernel theorem) that infinite-dimensional Banach spaces lack (for more details, see the properties, sufficient conditions, and characterizations given in the article Nuclear space). It is in this sense that nuclear spaces are an "alternative generalization" of finite-dimensional spaces. Also, as a general rule, in practice most "naturally occurring" TVSs are usually either Banach spaces or nuclear space. Typically, most TVSs that are associated with smoothness (i.e. ''infinite'' differentiability, such as and ) end up being nuclear TVSs while TVSs associated with ''finite'' continuous differentiability (such as with compact and ) often end up being non-nuclear spaces, such as Banach spaces.]
;Relationships between spaces
Using the universal property of direct limit
s and the fact that the natural inclusions
are all topological embedding
, one may show that all of the maps
are also topological embeddings. Said differently, the topology on
is identical to the subspace topology
that it inherits from
where recall that
's topology was ''defined'' to be the subspace topology induced on it by
In particular, both
induces the same subspace topology on
However, this does ''not'' imply that the canonical LF topology on
is equal to the subspace topology induced on
; these two topologies on
are in fact ''never'' equal to each other since the canonical LF topology is ''never'' metrizable while the subspace topology induced on it by
is metrizable (since recall that
is metrizable). The canonical LF topology on
is actually ''strictly finer
'' than the subspace topology that it inherits from
(thus the natural inclusion
is continuous but ''not'' a topological embedding
Indeed, the canonical LF topology is so fine
denotes some linear map that is a "natural inclusion" (such as
or other maps discussed below) then this map will typically be continuous, which as is shown below, is ultimately the reason why locally integrable functions, Radon measure
s, etc. all induce distributions (via the transpose of such a "natural inclusion"). Said differently, the reason why there are so many different ways of defining distributions from other spaces ultimately stems from how very fine the canonical LF topology is. Moreover, since distributions are just continuous linear functionals on
the fine nature of the canonical LF topology means that more linear functionals on
end up being continuous ("more" means as compared to a coarser topology that we could have placed on
such as for instance, the subspace topology induced by some
which although it would have made
metrizable, it would have also resulted in fewer linear functionals on
being continuous and thus there would have been fewer distributions; moreover, this particular coarser topology also has the disadvantage of not making
into a complete TVS
* The differentiation map
is a surjective continuous linear operator.
* The bilinear multiplication map
is ''not'' continuous; it is however, hypocontinuous
As discussed earlier, continuous linear functionals
are known as distributions on . Thus the set of all distributions on is the continuous dual space
which when endowed with the strong dual topology
is denoted by
We have the canonical duality pairing
between a distribution on and a test function
which is denoted using angle brackets
One interprets this notation as the distribution acting on the test function
to give a scalar, or symmetrically as the test function
acting on the distribution .
;Characterizations of distributions
Proposition. If is a linear functional
then the following are equivalent:
# is a distribution;
# (''definition'') is continuous
# is continuous
at the origin;
# is uniformly continuous
# is a bounded operator
# is sequentially continuous
#* explicitly, for every sequence
that converges in
[Even though the topology of is not metrizable, a linear functional on is continuous if and only if it is sequentially continuous.]
# is sequentially continuous
at the origin; in other words, maps null sequences
to null sequences;
#* explicitly, for every sequence
that converges in
to the origin (such a sequence is called a ''null sequence''),
#* a ''null sequence'' is by definition a sequence that converges to the origin;
# maps null sequences to bounded subsets;
#* explicitly, for every sequence
that converges in
to the origin, the sequence
# maps Mackey convergence null sequences
to bounded subsets;
#* explicitly, for every Mackey convergent null sequence
#* a sequence is said to be ''Mackey convergent
to '' if there exists a divergent sequence of positive real number such that the sequence is bounded; every sequence that is Mackey convergent to necessarily converges to the origin (in the usual sense);
# The kernel of is a closed subspace of
# The graph of is a closed;
# There exists a continuous seminorm on
# There exists a constant , a collection of continuous seminorms,
that defines the canonical LF topology of
and a finite subset
[If is also directed under the usual function comparison then we can take the finite collection to consist of a single element.]
# For every compact subset
there exist constants
such that for all
# For every compact subset
there exist constants
such that for all
with support contained in
# For any compact subset
and any sequence
converges uniformly to zero for all multi-indices
# Any of the ''three'' statements immediately above (i.e. statements 14, 15, and 16) but with the additional requirement that compact set belongs to .
Topology on the space of distributions
The topology of uniform convergence on bounded subsets is also called ''the strong dual topology
[In functional analysis, the strong dual topology is often the "standard" or "default" topology placed on the continuous dual space where if is a normed space then this strong dual topology is the same as the usual norm-induced topology on ]
This topology is chosen because it is with this topology that
becomes a nuclear Montel space
and it is with this topology that the kernels theorem of Schwartz
holds. No matter what dual topology is placed on
[Technically, the topology must be coarser than the strong dual topology and also simultaneously be finer that the weak* topology.]
a ''sequence'' of distributions converges in this topology if and only if it converges pointwise (although this need not be true of a net
). No matter which topology is chosen,
will be a non-metrizable
, locally convex topological vector space
. The space
and has the strong Pytkeev property
[Gabriyelyan, S.S. Kakol J., and·Leiderman, A]
"The strong Pitkeev property for topological groups and topological vector spaces"
/ref> but it is neither a k-space
nor a sequential space, which in particular implies that it is not metrizable and also that its topology can ''not'' be defined using only sequences.
;Topological vector space categories
The canonical LF topology makes into a complete distinguished strict LF-space (and a strict LB-space if and only if ), which implies that is a meager subset of itself. Furthermore, as well as its strong dual space, is a complete Hausdorff locally convex barrelled bornological Mackey space. The strong dual of is a Fréchet space if and only if so in particular, the strong dual of which is the space of distributions on , is ''not'' metrizable (note that the weak-* topology on also is not metrizable and moreover, it further lacks almost all of the nice properties that the strong dual topology gives ).
The three spaces and the Schwartz space as well as the strong duals of each of these three spaces, are complete nuclear Montel bornological spaces, which implies that all six of these locally convex spaces are also paracompact
reflexive barrelled Mackey spaces. The spaces and are both distinguished Fréchet spaces. Moreover, both and are Schwartz TVSs.
;Convergent sequences and their insufficiency to describe topologies
The strong dual spaces of and are sequential spaces but not Fréchet-Urysohn spaces.
Moreover, neither the space of test functions nor its strong dual is a sequential space (not even an Ascoli space), [Gabriyelyan, Saa]
"Topological properties of Strict LF-spaces and strong duals of Montel Strict LF-spaces"
[T. Shirai, Sur les Topologies des Espaces de L. Schwartz, Proc. Japan Acad. 35 (1959), 31-36.] which in particular implies that their topologies can ''not'' be defined entirely in terms of convergent sequences.
A sequence in converges in if and only if there exists some such that contains this sequence and this sequence converges in ; equivalently, it converges if and only if the following two conditions hold:
# There is a compact set containing the supports of all
# For each multi-index , the sequence of partial derivatives tends uniformly to
Neither the space nor its strong dual is a sequential space, and consequently, their topologies can ''not'' be defined entirely in terms of convergent sequences. For this reason, the above characterization of when a sequence converges is ''not'' enough to define the canonical LF topology on The same can be said of the strong dual topology on
;What sequences do characterize
Nevertheless, sequences do characterize many important properties, as we now discuss. It is known that in the dual space of any Montel space, a sequence converges in the strong dual topology if and only if it converges in the weak* topology, which in particular, is the reason why a sequence of distributions converges (in the strong dual topology) if and only if it converges pointwise (this leads many authors to use pointwise convergence to actually ''define'' the convergence of a sequence of distributions; this is fine for sequences but it does ''not'' extend to the convergence of nets of distributions since a net may converge pointwise but fail to convergence in the strong dual topology).
Sequences characterize continuity of linear maps valued in locally convex space. Suppose is a locally convex bornological space (such as any of the six TVSs mentioned earlier). Then a linear map into a locally convex space is continuous if and only if it maps null sequences [A ''null sequence'' is a sequence that converges to the origin.] in to bounded subsets of . [Recall that a linear map is bounded if and only if it maps null sequences to bounded sequences.] More generally, such a linear map is continuous if and only if it maps Mackey convergent null sequences [A sequence is said to be ''Mackey convergent to in '' if there exists a divergent sequence of positive real number such that is a bounded set in ] to bounded subsets of So in particular, if a linear map into a locally convex space is sequentially continuous at the origin then it is continuous. However, this does ''not'' necessarily extend to non-linear maps and/or to maps valued in topological spaces that are not locally convex TVSs.
For every is sequentially dense in Furthermore, is a sequentially dense subset of (with its strong dual topology) and also a sequentially dense subset of the strong dual space of
;Sequences of distributions
A sequence of distributions converges with respect to the weak-* topology on to a distribution if and only if
for every test function For example, if is the function
and is the distribution corresponding to then
as , so in Thus, for large , the function can be regarded as an approximation of the Dirac delta distribution.
* The strong dual space of is TVS isomorphic to via the canonical TVS-isomorphism defined by sending to ''value at '' (i.e. to the linear functional on defined by sending to );
* On any bounded subset of the weak and strong subspace topologies coincide; the same is true for ;
* Every weakly convergent sequence in is strongly convergent (although this does not extend to nets).
Localization of distributions
There is no way to define the value of a distribution in at a particular point of . However, as is the case with functions, distributions on restrict to give distributions on open subsets of . Furthermore, distributions are ''locally determined'' in the sense that a distribution on all of can be assembled from a distribution on an open cover of satisfying some compatibility conditions on the overlaps. Such a structure is known as a sheaf.
Restrictions to an open subset
Let and be open subsets of with . Let be the operator which ''extends by zero'' a given smooth function compactly supported in to a smooth function compactly supported in the larger set . The transpose of is called the restriction mapping and is denoted by
The map is a continuous injection where if then it is ''not'' a topological embedding and its range is ''not'' dense in which implies that this map's transpose is neither injective nor surjective and that the topology that transfers from onto its image is strictly finer than the subspace topology that induces on this same set. A distribution is said to be ''extendible to '' if it belongs to the range of the transpose of and it is called ''extendible'' if it is extendable to
For any distribution the restriction is a distribution in defined by:
Unless , the restriction to is neither injective nor surjective. Lack of surjectivity follows since distributions can blow up towards the boundary of . For instance, if and , then the distribution
is in but admits no extension to
Gluing and distributions that vanish in a set
Let be an open subset of . is said to ''vanish in '' if for all such that we have vanishes in if and only if the restriction of to is equal to 0, or equivalently, if and only if lies in the kernel of the restriction map .
:Corollary. Let be a collection of open subsets of and let if and only if for each the restriction of to is equal to 0.
:Corollary. The union of all open subsets of in which a distribution vanishes is an open subset of in which vanishes.
Support of a distribution
This last corollary implies that for every distribution on , there exists a unique largest subset of such that vanishes in (and does not vanish in any open subset of that is not contained in ); the complement in of this unique largest open subset is called ''the support of ''. Thus
If is a locally integrable function on and if is its associated distribution, then the support of is the smallest closed subset of in the complement of which is almost everywhere equal to 0. If is continuous, then the support of is equal to the closure of the set of points in at which does not vanish. The support of the distribution associated with the Dirac measure at a point is the set If the support of a test function does not intersect the support of a distribution then . A distribution is 0 if and only if its support is empty. If is identically 1 on some open set containing the support of a distribution then . If the support of a distribution is compact then it has finite order and furthermore, there is a constant ''C'' and a non-negative integer ''N'' such that:
If has compact support then it has a unique extension to a continuous linear functional on ; this functional can be defined by where is any function that is identically 1 on an open set containing the support of .
If and then and Thus, distributions with support in a given subset form a vector subspace of ; such a subspace is weakly closed in if and only if ''A'' is closed in . Furthermore, if is a differential operator in , then for all distributions on and all we have and
Distributions with compact support
;Support in a point set and Dirac measures
For any let denote the distribution induced by the Dirac measure at ''x''. For any and distribution the support of is contained in if and only if is a finite linear combination of derivatives of the Dirac measure at If in addition the order of is then there exist constants such that:
Said differently, if has support at a single point then is in fact a finite linear combination of distributional derivatives of the function at . That is, there exists an integer and complex constants such that
where is the translation operator.
;Distribution with compact support
;Distributions of finite order with support in an open subset
Global structure of distributions
The formal definition of distributions exhibits them as a subspace of a very large space, namely the topological dual of (or the Schwartz space 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.
;Distributions as sheafs
Decomposition of distributions as sums of derivatives of continuous functions
By combining the above results, one may express any distribution on 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 . In other words for arbitrary we can write:
where are finite sets of multi-indices and the functions are continuous.
Note that the infinite sum above is well-defined as a distribution. The value of for a given can be computed using the finitely many that intersect the support of
Operations on distributions
Many operations which are defined on smooth functions with compact support can also be defined for distributions. In general, if is a linear map which is continuous with respect to the weak topology, then it is possible to extend to a map by passing to the limit.
[This approach works for non-linear mappings as well, provided they are assumed to be uniformly continuous.]
Preliminaries: Transpose of a linear operator
Operations on distributions and spaces of distributions are often defined by means of the transpose of a linear operator because it provides a unified approach that the many definitions in the theory of distributions and because of its many well-known topological properties. In general the transpose of a continuous linear map is the linear map defined by or equivalently, it is the unique map satisfying for all and all Since is continuous, the transpose 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 be a continuous linear map. Then by definition, the transpose of is the unique linear operator that satisfies:
: for all and all
However, since the image of is dense in it is sufficient that the above equality hold for all distributions of the form where Explicitly, this means that the above condition holds if and only if the condition below holds:
: for all
Differentiation of distributions
Let is the partial derivative operator In order to extend we compute its transpose:
Therefore Therefore the partial derivative of with respect to the coordinate is defined by the formula
With this definition, every distribution is infinitely differentiable, and the derivative in the direction is a linear operator on
More generally, if is an arbitrary multi-index, then the partial derivative of the distribution is defined by
Differentiation of distributions is a continuous operator on this is an important and desirable property that is not shared by most other notions of differentiation.
If is a distribution in then
where is the derivative of and is translation by ; thus the derivative of may be viewed as a limit of quotients.
Differential operators acting on smooth functions
A linear differential operator in with smooth coefficients acts on the space of smooth functions on Given we would like to define a continuous linear map, that extends the action of on to distributions on In other words we would like to define such that the following diagram commutes:
Where the vertical maps are given by assigning its canonical distribution which is defined by: for all With this notation the diagram commuting is equivalent to:
In order to find we consider the transpose of the continuous induced map defined by As discussed above, for any the transpose may be calculated by:
For the last line we used integration by parts combined with the fact that and therefore all the functions have compact support.
[For example let and take to be the ordinary derivative for functions of one real variable and assume the support of to be contained in the finite interval then since
where the last equality is because ] Continuing the calculation above we have for all
Define ''the formal transpose of '' which will be denoted by to avoid confusion with the transpose map, to be the following differential operator on :
The computations above have shown that:
:Lemma. Let be a linear differential operator with smooth coefficients in Then for all we have
:which is equivalent to:
The Lemma combined with the fact that the formal transpose of the formal transpose is the original differential operator, i.e. enables us to arrive at the correct definition: the formal transpose induces the (continuous) canonical linear operator defined by We claim that the transpose of this map, can be taken as To see this, for every , compute its action on a distribution of the form with :
We call the continuous linear operator ''the differential operator on distributions extending ''. Its action on an arbitrary distribution is defined via:
If converges to then for every multi-index converges to
Multiplication of distributions by smooth functions
A differential operator of order 0 is just multiplication by a smooth function. And conversely, if is a smooth function then is a differential operator of order 0, whose formal transpose is itself (i.e. ). The induced differential operator maps a distribution to a distribution denoted by We have thus defined the multiplication of a distribution by a smooth function.
We now give an alternative presentation of multiplication by a smooth function. If is a smooth function and is a distribution on , then the product ''mT'' is defined by
This definition coincides with the transpose definition since if is the operator of multiplication by the function (i.e., ), then
Under multiplication by smooth functions, is a module over the ring With this definition of multiplication by a smooth function, the ordinary product rule of calculus remains valid. However, a number of unusual identities also arise. For example, if is the Dirac delta distribution on , then , and if is the derivative of the delta distribution, then
The bilinear multiplication map given by is ''not'' continuous; it is however, hypocontinuous.
Example. For any distribution , the product of with the function that is identically on is equal to .
Example. Suppose is a sequence of test functions on that converges to the constant function For any distribution on , the sequence converges to
If converges to and converges to then converges to
= Problem of multiplying distributions
It is easy to define the product of a distribution with a smooth function, or more generally the product of two distributions whose singular supports are disjoint. 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 p.v. is the distribution obtained by the Cauchy principal value
If is the Dirac delta distribution then
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 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) ''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 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, Martin Hairer proposed a consistent way of multiplying distributions with certain structure (regularity structures), 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.
Composition with a smooth function
Let be a distribution on Let be an open set in and . If is a submersion, it is possible to define
This is ''the composition of the distribution with '', and is also called ''the pullback of along '', sometimes written
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 be a submersion is equivalent to the requirement that the Jacobian derivative of is a surjective linear map for every . A necessary (but not sufficient) condition for extending to distributions is that be an open mapping. The inverse function theorem ensures that a submersion satisfies this condition.
If is a submersion, then is defined on distributions by finding the transpose map. Uniqueness of this extension is guaranteed since is a continuous linear operator on Existence, however, requires using the change of variables formula, the inverse function theorem (locally) and a partition of unity argument.
In the special case when is a diffeomorphism from an open subset of onto an open subset of change of variables under the integral gives
In this particular case, then, is defined by the transpose formula:
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 and are functions on then we denote by ''the convolution of and '', defined at to be the integral
provided that the integral exists. If are such that 1/''r'' = (1/''p'') + (1/''q'') - 1 then for any functions and we have and If and are continuous functions on at least one of which has compact support, then and if then the value of on do ''not'' depend on the values of outside of the Minkowski sum
Importantly, if has compact support then for any the convolution map is continuous when considered as the map or as the map
;Translation and symmetry
Given the translation operator sends to defined by This can be extended by the transpose to distributions in the following way: given a distribution , ''the translation of by '' is the distribution defined by
Given define the function by Given a distribution , let be the distribution defined by The operator is called ''the symmetry with respect to the origin''.
Convolution of a test function with a distribution
Convolution with defines a linear map:
which is continuous with respect to the canonical LF space topology on
Convolution of with a distribution can be defined by taking the transpose of ''Cf'' relative to the duality pairing of with the space of distributions. If then by Fubini's theorem
Extending by continuity, the convolution of with a distribution is defined by
An alternative way to define the convolution of a test function and a distribution is to use the translation operator . The convolution of the compactly supported function and the distribution is then the function defined for each by
It can be shown that the convolution of a smooth, compactly supported function and a distribution is a smooth function. If the distribution has compact support then if is a polynomial (resp. an exponential function, an analytic function, the restriction of an entire analytic function on to the restriction of an entire function of exponential type in to ) then the same is true of If the distribution has compact support as well, then is a compactly supported function, and the Titchmarsh convolution theorem implies that
where ''ch'' denotes the convex hull and supp denotes the support.
Convolution of a smooth function with a distribution
Let and and assume that at least one of and has compact support. The ''convolution of and '', denoted by or by is the smooth function:
satisfying for all :
If is a distribution then the map is continuous as a map where if in addition has compact support then it is also continuous as the map and continuous as the map
If is a continuous linear map such that for all and all then there exists a distribution such that for all
Example. Let ''H'' be the Heaviside function on . For any
Let be the Dirac measure at 0 and its derivative as a distribution. Then and Importantly, the associative law fails to hold:
Convolution of distributions
It is also possible to define the convolution of two distributions and on provided one of them has compact support. Informally, in order to define where has compact support, the idea is to extend the definition of the convolution to a linear operation on distributions so that the associativity formula
continues to hold for all test functions
It is also possible to provide a more explicit characterization of the convolution of distributions. Suppose that and are distributions and that has compact support. Then the linear maps
are continuous. The transposes of these maps,
are consequently continuous and one may show that
This common value is called ''the convolution of and '' and it is a distribution that is denoted by or It satisfies If and are two distributions, at least one of which has compact support, then for any If is a distribution in and if is a Dirac measure then
Suppose that it is that has compact support. For consider the function
It can be readily shown that this defines a smooth function of , which moreover has compact support. The convolution of and is defined by
This generalizes the classical notion of convolution of functions and is compatible with differentiation in the following sense: for every multi-index ,
The convolution of a finite number of distributions, all of which (except possibly one) have compact support, is associative.
This definition of convolution remains valid under less restrictive assumptions about and .
The convolution of distributions with compact support induces a continuous bilinear map defined by where denotes the space of distributions with compact support. However, the convolution map as a function is ''not'' continuous although it is separately continuous. The convolution maps and given by both ''fail'' to be continuous. Each of these non-continuous maps is, however, separately continuous and hypocontinuous.
Convolution versus multiplication
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 be a rapidly decreasing tempered distribution or, equivalently, be an ordinary (slowly growing, smooth) function within the space of tempered distributions and let be the normalized (unitary, ordinary frequency) Fourier transform then, according to ,
hold within the space of tempered distributions. In particular, these equations become the Poisson Summation Formula if is the Dirac Comb. The space of all rapidly decreasing tempered distributions is also called the space of ''convolution operators'' and the space of all ordinary functions within the space of tempered distributions is also called the space of ''multiplication operators'' More generally, and
A particular case is the Paley-Wiener-Schwartz Theorem which states that and This is because and In other words, compactly supported tempered distributions belong to the space of ''convolution operators'' and
Paley-Wiener functions better known as bandlimited functions, belong to the space of ''multiplication operators''
For example, let be the Dirac comb and be the Dirac delta then is the function that is constantly one and both equations yield the Dirac comb identity. Another example is to let be the Dirac comb and be the rectangular function then is the sinc function and both equations yield the Classical Sampling Theorem for suitable functions. More generally, if is the Dirac comb and is a smooth window function (Schwartz function), e.g. the Gaussian, then 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.
Tensor product of distributions
Let and be open sets. Assume all vector spaces to be over the field where or For we define the following family of functions:
Given and we define the following functions:
Note that and Now we define the following continuous linear maps associated to and :
Moreover if either (resp. ) has compact support then it also induces a continuous linear map of (resp. ).
Definition. ''The tensor product of and '' denoted by or is a distribution in and is defined by:
Schwartz kernel theorem
The tensor product defines a bilinear map
the span of the range of this map is a dense subspace of its codomain. Furthermore, Moreover induces continuous bilinear maps:
where denotes the space of distributions with compact support and is the Schwartz space of rapidly decreasing functions.
This result does not hold for Hilbert spaces such as and its dual space. Why does such a result hold for the space of distributions and test functions but not for other "nice" spaces like the Hilbert space ? This question led Alexander Grothendieck to discover nuclear spaces, nuclear maps, and the injective tensor product. He ultimately showed that it is precisely because is a nuclear space that the Schwartz kernel theorem holds.
Spaces of distributions
For all and all , all of the following canonical injections are continuous and have a range that is dense in their codomain:
where the topologies on () are defined as direct limits of the spaces in a manner analogous to how the topologies on 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 the codomain. Indeed, is even sequentially dense in every All of the canonical injections () are continuous and the range of this injection is dense in the codomain if and only if (here has its usual norm topology).
Suppose that is one of the spaces () or () or (). Since the canonical injection is a continuous injection whose image is dense in the codomain, the transpose is a continuous injection. This transpose thus allows us to identify with a certain vector subspace of the space of distributions. This transpose map is not necessarily a TVS-embedding so that topology that this map transfers to the image is finer than the subspace topology that this space inherits from
A linear subspace of carrying a locally convex topology that is finer than the subspace topology induced by is called ''a space of distributions''.
Almost all of the spaces of distributions mentioned in this article arise in this way (e.g. tempered distribution, restrictions, distributions of order some integer, distributions induced by a positive Radon measure, distributions induced by an -function, etc.) and any representation theorem about the dual space of may, through the transpose be transferred directly to elements of the space
The natural inclusion is a continuous injection whose image is dense in its codomain, so the transpose is also a continuous injection.
Note that the continuous dual space can be identified as the space of Radon measures, where there is a one-to-one correspondence between the continuous linear functionals and integral with respect to a Radon measure; that is,
* if then there exists a Radon measure on such that for all and
* if is a Radon measure on then the linear functional on defined by is continuous.
Through the injection every Radon measure becomes a distribution on . If is a locally integrable function on then the distribution 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 functions in :
:Theorem. Suppose is a Radon measure, is a neighborhood of the support of , and There exists is a family of locally functions in such that
:and for very
;Positive Radon measures
A linear function on a space of functions is called ''positive'' if whenever a function that belongs to the domain of is non-negative (i.e. is real-valued and ) then One may show that every positive linear functional on is necessarily continuous (i.e. necessarily a Radon measure).Note that Lebesgue measure is an example of a positive Radon measure.
Locally integrable functions as distributions
One particularly important class of Radon measures are those that are induced locally integrable functions. The function is called ''locally integrable'' if it is Lebesgue integrable over every compact subset of .
[For more information on such class of functions, see the entry on locally integrable functions.] This is a large class of functions which includes all continuous functions and all ''Lp'' functions. The topology on is defined in such a fashion that any locally integrable function yields a continuous linear functional on – that is, an element of – denoted here by , whose value on the test function is given by the Lebesgue integral:
Conventionally, one abuses notation by identifying with provided no confusion can arise, and thus the pairing between and is often written
If and are two locally integrable functions, then the associated distributions and are equal to the same element of if and only if and are equal almost everywhere (see, for instance, ). In a similar manner, every Radon measure on defines an element of whose value on the test function is As above, it is conventional to abuse notation and write the pairing between a Radon measure and a test function as 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.
;Test functions as distributions
The test functions are themselves locally integrable, and so define distributions. The space of test functions is sequentially dense in with respect to the strong topology on This means that for any there is a sequence of test functions, that converges to (in its strong dual topology) when considered as a sequence of distributions. Or equivalently,
Furthermore, is also sequentially dense in the strong dual space of
Distributions with compact support
The natural inclusion is a continuous injection whose image is dense in its codomain, so the transpose is also a continuous injection. Thus the image of the transpose, denoted by forms a space of distributions when it is endowed with the strong dual topology of (transferred to it via the transpose map so the topology of is finer than the subspace topology that this set inherits from ).
The elements of can be identified as the space of distributions with compact support. Explicitly, if is a distribution on then the following are equivalent,
* the support of is compact;
* the restriction of to when that space is equipped with the subspace topology inherited from (a coarser topology than the canonical LF topology), is continuous;
* there is a compact subset of such that for every test function whose support is completely outside of , we have
Compactly supported distributions define continuous linear functionals on the space ; recall that the topology on is defined such that a sequence of test functions converges to 0 if and only if all derivatives of converge uniformly to 0 on every compact subset of . 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 to
Distributions of finite order
Let The natural inclusion is a continuous injection whose image is dense in its codomain, so the transpose is also a continuous injection. Consequently, the image of denoted by forms a space of distributions when it is endowed with the strong dual topology of (transferred to it via the transpose map so 's topology is finer than the subspace topology that this set inherits from ). The elements of are ''the distributions of order ''. The distributions of order ≤ 0, which are also called ''distributions of order '', are exactly the distributions that are Radon measures (described above).
For a ''distribution of order '' is a distribution of order that is not a distribution of order .
A distribution is said to be of ''finite order'' if there is some integer such that it is a distribution of order , and the set of distributions of finite order is denoted by Note that if then so that is a vector subspace of and furthermore, if and only if
;Structure of distributions of finite order
Every distribution with compact support in is a distribution of finite order. Indeed, every distribution in is ''locally'' a distribution of finite order, in the following sense: If is an open and relatively compact subset of and if is the restriction mapping from to , then the image of under is contained in
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:
:Theorem. Suppose has finite order and Given any open subset of containing the support of , there is a family of Radon measures in , such that for very and
Example. (Distributions of infinite order) Let and for every test function let
Then is a distribution of infinite order on . Moreover, can not be extended to a distribution on ; that is, there exists no distribution on such that the restriction of to is equal to .
Tempered distributions and Fourier transform
Defined below are the ''tempered distributions'', which form a subspace of the space of distributions on This is a proper subspace: while every tempered distribution is a distribution and an element of 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
The Schwartz space, is the space of all smooth functions that are rapidly decreasing at infinity along with all partial derivatives. Thus is in the Schwartz space provided that any derivative of multiplied with any power of , converges to 0 as . These functions form a complete TVS with a suitably defined family of seminorms. More precisely, for any multi-indices and define:
Then is in the Schwartz space if all the values satisfy:
The family of seminorms 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:
Otherwise, one can define a norm on via
The Schwartz space is a Fréchet space (i.e. a complete metrizable locally convex space). Because the Fourier transform changes into multiplication by and vice versa, this symmetry implies that the Fourier transform of a Schwartz function is also a Schwartz function.
A sequence in converges to 0 in if and only if the functions converge to 0 uniformly in the whole of which implies that such a sequence must converge to zero in
is dense in The subset of all analytic Schwartz functions is dense in as well.
The Schwartz space is nuclear and the tensor product of two maps induces a canonical surjective TVS-isomorphisms
where represents the completion of the injective tensor product (which in this case is the identical to the completion of the projective tensor product).
The natural inclusion is a continuous injection whose image is dense in its codomain, so the transpose is also a continuous injection. Thus, the image of the transpose map, denoted by forms a space of distributions when it is endowed with the strong dual topology of (transferred to it via the transpose map so the topology of is finer than the subspace topology that this set inherits from ).
The space is called the space of ''>tempered distributions'' is it is the continuous dual of the Schwartz space. Equivalently, a distribution is a tempered distribution if and only if
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 for are tempered distributions.
The ''tempered distributions'' can also be characterized as ''slowly growing'', meaning that each derivative of grows at most as fast as some polynomial. This characterization is dual to the ''rapidly falling'' behaviour of the derivatives of a function in the Schwartz space, where each derivative of decays faster than every inverse power of . An example of a rapidly falling function is for any positive , , .
To study the Fourier transform, it is best to consider complex-valued test functions and complex-linear distributions. The ordinary continuous Fourier transform is a TVS-automorphism of the Schwartz space, and ''the Fourier transform'' is defined to be its transpose which (abusing notation) will again be denoted by . So the Fourier transform of the tempered distribution is defined by for every Schwartz function . 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
and also with convolution: if is a tempered distribution and is a ''slowly increasing'' smooth function on is again a tempered distribution and
is the convolution of and . In particular, the Fourier transform of the constant function equal to 1 is the distribution.
;Expressing tempered distributions as sums of derivatives
If is a tempered distribution, then there exists a constant , and positive integers and such that for all Schwartz functions
This estimate along with some techniques from functional analysis can be used to show that there is a continuous slowly increasing function and a multi-index such that
Restriction of distributions to compact sets
If then for any compact set there exists a continuous function compactly supported in (possibly on a larger set than itself) and a multi-index such that on
Using holomorphic functions as test functions
The success of the theory led to 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.
* Colombeau algebra
* Current (mathematics)
* Distribution (number theory)
* Distribution on a linear algebraic group
* Gelfand triple
* Generalized function
* Homogeneous distribution
* Laplacian of the indicator
* Limit of a distribution
* Linear form
* Malgrange–Ehrenpreis theorem
* Pseudodifferential operator
* Riesz representation theorem
* Vague topology
* Weak solution
*M. J. Lighthill (1959). ''Introduction to Fourier Analysis and Generalised Functions''. Cambridge University Press. (requires very little knowledge of analysis; defines distributions as limits of sequences of functions under integrals)
*V.S. Vladimirov (2002). ''Methods of the theory of generalized functions''. Taylor & Francis.
Category:Generalizations of the derivative