International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 31-34   | 1 | 2 |

## Section 1.3.2.3.9. Operations on distributions

G. Bricognea

aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

#### 1.3.2.3.9. Operations on distributions

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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.

#### 1.3.2.3.9.1. Differentiation

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• (a) Definition and elementary properties

If T is a distribution on , its partial derivative with respect to is defined by

for all . This does define a distribution, because the partial differentiations are continuous for the topology of .

Suppose that with f a locally integrable function such that exists and is almost everywhere continuous. Then integration by parts along the axis gives the integrated term vanishes, since ϕ has compact support, showing that .

The test functions are infinitely differentiable. Therefore, transpositions like that used to define may be repeated, so that any distribution is infinitely differentiable. For instance, where Δ is the Laplacian operator. The derivatives of Dirac's δ distribution are

It is remarkable that differentiation is a continuous operation for the topology on : if a sequence of distributions converges to distribution T, then the sequence of derivatives converges to for any multi-index p, since as An analogous statement holds for series: any convergent series of distributions may be differentiated termwise to all orders. This illustrates how robust' the constructs of distribution theory are in comparison with those of ordinary function theory, where similar statements are notoriously untrue.

• (b) Differentiation under the duality bracket

Limiting processes and differentiation may also be carried out under the duality bracket as under the integral sign with ordinary functions. Let the function depend on a parameter and a vector in such a way that all functions be in for all . Let be a distribution, let and let be given parameter value. Suppose that, as λ runs through a small enough neighbourhood of ,

 (i) all the have their supports in a fixed compact subset K of ; (ii) all the derivatives have a partial derivative with respect to λ which is continuous with respect to x and λ.

Under these hypotheses, is differentiable (in the usual sense) with respect to λ near , and its derivative may be obtained by differentiation under the sign':

• (c) Effect of discontinuities

When a function f or its derivatives are no longer continuous, the derivatives of the associated distribution may no longer coincide with the distributions associated to the functions .

In dimension 1, the simplest example is Heaviside's unit step function : Hence , a result long used heuristically' by electrical engineers [see also Dirac (1958)].

Let f be infinitely differentiable for and but have discontinuous derivatives at [ being f itself] with jumps . Consider the functions: The are continuous, their derivatives are continuous almost everywhere [which implies that and almost everywhere]. This yields immediately: Thus the distributional derivatives' differ from the usual functional derivatives by singular terms associated with discontinuities.

In dimension n, let f be infinitely differentiable everywhere except on a smooth hypersurface S, across which its partial derivatives show discontinuities. Let and denote the discontinuities of f and its normal derivative across S (both and are functions of position on S), and let and be defined by Integration by parts shows that where is the angle between the axis and the normal to S along which the jump occurs, and that the Laplacian of is given by The latter result is a statement of Green's theorem in terms of distributions. It will be used in Section 1.3.4.4.3.5 to calculate the Fourier transform of the indicator function of a molecular envelope.

#### 1.3.2.3.9.2. Integration of distributions in dimension 1

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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.

#### 1.3.2.3.9.3. Multiplication of distributions by functions

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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 multi-index p

#### 1.3.2.3.9.4. Division of distributions by functions

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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)].

#### 1.3.2.3.9.5. Transformation of coordinates

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Let σ be a smooth non-singular 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 non-singular 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 .

#### 1.3.2.3.9.6. Tensor product of distributions

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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.

#### 1.3.2.3.9.7. Convolution of distributions

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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:

 (i) : is the unit convolution; (ii) : translation is a convolution with the corresponding translate of δ; (iii) : differentiation is a convolution with the corresponding derivative of δ; (iv) translates or derivatives of a convolution may be obtained by translating or differentiating any one of the factors: convolution commutes' with translation and differentiation, a property used in Section 1.3.4.4.7.7 to speed up least-squares model refinement for macromolecules.

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 integro-differential equations with constant coefficients by turning them into convolution equations, then using factorization methods for convolution algebras (Schwartz, 1965).

### References

Bracewell, R. N. (1986). The Fourier transform and its applications, 2nd ed., revised. New York: McGraw-Hill.Google Scholar
Carslaw, H. S. & Jaeger, J. C. (1948). Operational methods in applied mathematics. Oxford University Press.Google Scholar
Churchill, R. V. (1958). Operational mathematics, 2nd ed. New York: McGraw-Hill.Google Scholar
Dirac, P. A. M. (1958). The principles of quantum mechanics, 4th ed. Oxford: Clarendon Press.Google Scholar
Erdélyi, A. (1962). Operational calculus and generalized functions. New York: Holt, Rinehart & Winston.Google Scholar
Hörmander, L. (1963). Linear partial differential operators. Berlin: Springer-Verlag.Google Scholar
Moore, D. H. (1971). Heaviside operational calculus. An elementary foundation. New York: American Elsevier.Google Scholar