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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 34-35   | 1 | 2 |

## Section 1.3.2.3.9.7. Convolution of distributions

G. Bricognea

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#### 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.
Carslaw, H. S. & Jaeger, J. C. (1948). Operational Methods in Applied Mathematics. Oxford University Press.
Churchill, R. V. (1958). Operational Mathematics, 2nd ed. New York: McGraw-Hill.
Erdélyi, A. (1962). Operational Calculus and Generalized Functions. New York: Holt, Rinehart & Winston.
Moore, D. H. (1971). Heaviside Operational Calculus. An Elementary Foundation. New York: American Elsevier.
Schwartz, L. (1965). Mathematics for the Physical Sciences. Paris: Hermann, and Reading: Addison-Wesley.
Van der Pol, B. & Bremmer, H. (1955). Operational Calculus, 2nd ed. Cambridge University Press.