International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 3134

As a general rule, the definitions are chosen so that the operations coincide with those on functions whenever a distribution is associated to a function.
Most definitions consist in transferring to a distribution T an operation which is well defined on by `transposing' it in the duality product ; this procedure will map T to a new distribution provided the original operation maps continuously into itself.
The reverse operation from differentiation, namely calculating the `indefinite integral' of a distribution S, consists in finding a distribution T such that .
For all such that with , we must have This condition defines T in a `hyperplane' of , whose equation reflects the fact that ψ has compact support.
To specify T in the whole of , it suffices to specify the value of where is such that : then any may be written uniquely as with and T is defined by The freedom in the choice of means that T is defined up to an additive constant.
The product of a distribution T on by a function α over will be defined by transposition: In order that be a distribution, the mapping must send continuously into itself; hence the multipliers α must be infinitely differentiable. The product of two general distributions cannot be defined. The need for a careful treatment of multipliers of distributions will become clear when it is later shown (Section 1.3.2.5.8) that the Fourier transformation turns convolutions into multiplications and vice versa.
If T is a distribution of order m, then α needs only have continuous derivatives up to order m. For instance, δ is a distribution of order zero, and is a distribution provided α is continuous; this relation is of fundamental importance in the theory of sampling and of the properties of the Fourier transformation related to sampling (Sections 1.3.2.6.4, 1.3.2.6.6). More generally, is a distribution of order , and the following formula holds for all with :
The derivative of a product is easily shown to be and generally for any multiindex p
Given a distribution S on and an infinitely differentiable multiplier function α, the division problem consists in finding a distribution T such that .
If α never vanishes, is the unique answer. If , and if α has only isolated zeros of finite order, it can be reduced to a collection of cases where the multiplier is , for which the general solution can be shown to be of the form where U is a particular solution of the division problem and the are arbitrary constants.
In dimension , the problem is much more difficult, but is of fundamental importance in the theory of linear partial differential equations, since the Fourier transformation turns the problem of solving these into a division problem for distributions [see Hörmander (1963)].
Let σ be a smooth nonsingular change of variables in , i.e. an infinitely differentiable mapping from an open subset Ω of to Ω′ in , whose Jacobian vanishes nowhere in Ω. By the implicit function theorem, the inverse mapping from Ω′ to Ω is well defined.
If f is a locally summable function on Ω, then the function defined by is a locally summable function on Ω′, and for any we may write: In terms of the associated distributions
This operation can be extended to an arbitrary distribution T by defining its image under coordinate transformation σ through which is well defined provided that σ is proper, i.e. that is compact whenever K is compact.
For instance, if is a translation by a vector a in , then ; is denoted by , and the translate of a distribution T is defined by
Let be a linear transformation defined by a nonsingular matrix A. Then , and This formula will be shown later (Sections 1.3.2.6.5, 1.3.4.2.1.1) to be the basis for the definition of the reciprocal lattice.
In particular, if , where I is the identity matrix, A is an inversion through a centre of symmetry at the origin, and denoting by we have: T is called an even distribution if , an odd distribution if .
If with , A is called a dilation and Writing symbolically δ as and as , we have: If and f is a function with isolated simple zeros , then in the same symbolic notation where each is analogous to a `Lorentz factor' at zero .
The purpose of this construction is to extend Fubini's theorem to distributions. Following Section 1.3.2.2.5, we may define the tensor product as the vector space of finite linear combinations of functions of the form where and .
Let and denote the distributions associated to f and g, respectively, the subscripts x and y acting as mnemonics for and . It follows from Fubini's theorem (Section 1.3.2.2.5) that , and hence defines a distribution over ; the rearrangement of integral signs gives for all . In particular, if with , then
This construction can be extended to general distributions and . Given any test function , let denote the map ; let denote the map ; and define the two functions and . Then, by the lemma on differentiation under the sign of Section 1.3.2.3.9.1, , and there exists a unique distribution such that is called the tensor product of S and T.
With the mnemonic introduced above, this definition reads identically to that given above for distributions associated to locally integrable functions:
The tensor product of distributions is associative: Derivatives may be calculated by The support of a tensor product is the Cartesian product of the supports of the two factors.
The convolution of two functions f and g on is defined by whenever the integral exists. This is the case when f and g are both in ; then is also in . Let S, T and W denote the distributions associated to f, g and respectively: a change of variable immediately shows that for any , Introducing the map σ from to defined by , the latter expression may be written: (where denotes the composition of mappings) or by a slight abuse of notation:
A difficulty arises in extending this definition to general distributions S and T because the mapping σ is not proper: if K is compact in , then is a cylinder with base K and generator the `second bisector' in . However, is defined whenever the intersection between Supp and is compact.
We may therefore define the convolution of two distributions S and T on by whenever the following support condition is fulfilled:
`the set is compact in for all K compact in '.
The latter condition is met, in particular, if S or T has compact support. The support of is easily seen to be contained in the closure of the vector sum
Convolution by a fixed distribution S is a continuous operation for the topology on : it maps convergent sequences to convergent sequences . Convolution is commutative: .
The convolution of p distributions with supports can be defined by whenever the following generalized support condition:
`the set is compact in for all K compact in '
is satisfied. It is then associative. Interesting examples of associativity failure, which can be traced back to violations of the support condition, may be found in Bracewell (1986, pp. 436–437).
It follows from previous definitions that, for all distributions , the following identities hold:

The latter property is frequently used for the purpose of regularization: if T is a distribution, α an infinitely differentiable function, and at least one of the two has compact support, then is an infinitely differentiable ordinary function. Since sequences of such functions α can be constructed which have compact support and converge to δ, it follows that any distribution T can be obtained as the limit of infinitely differentiable functions . In topological jargon: is `everywhere dense' in . A standard function in which is often used for such proofs is defined as follows: put with (so that θ is in and is normalized), and put
Another related result, also proved by convolution, is the structure theorem: the restriction of a distribution to a bounded open set Ω in is a derivative of finite order of a continuous function.
Properties (i) to (iv) are the basis of the symbolic or operational calculus (see Carslaw & Jaeger, 1948; Van der Pol & Bremmer, 1955; Churchill, 1958; Erdélyi, 1962; Moore, 1971) for solving integrodifferential equations with constant coefficients by turning them into convolution equations, then using factorization methods for convolution algebras (Schwartz, 1965).
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