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

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.2.3.9.7. Convolution of distributions

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The convolution [f * g] of two functions f and g on [{\bb R}^{n}] is defined by[(\,f * g) ({\bf x}) = {\textstyle\int\limits_{{\bb R}^{n}}} f({\bf y}) g({\bf x} - {\bf y}) \,\hbox{d}^{n}{\bf y} = {\textstyle\int\limits_{{\bb R}^{n}}} f({\bf x} - {\bf y}) g ({\bf y}) \,\hbox{d}^{n}{\bf y}]whenever the integral exists. This is the case when f and g are both in [L^{1} ({\bb R}^{n})]; then [f * g] is also in [L^{1} ({\bb R}^{n})]. Let S, T and W denote the distributions associated to f, g and [f * g,] respectively: a change of variable immediately shows that for any [\varphi \in {\scr D}({\bb R}^{n})],[\langle W, \varphi \rangle = {\textstyle\int\limits_{{\bb R}^{n} \times {\bb R}^{n}}} f({\bf x}) g({\bf y}) \varphi ({\bf x} + {\bf y}) \,\hbox{d}^{n}{\bf x} \,\hbox{d}^{n}{\bf y}.]Introducing the map σ from [{\bb R}^{n} \times {\bb R}^{n}] to [{\bb R}^{n}] defined by [\sigma ({\bf x}, {\bf y}) = {\bf x} + {\bf y}], the latter expression may be written:[\langle S_{\bf x} \otimes T_{\bf y}, \varphi \circ \sigma \rangle](where [\circ] denotes the composition of mappings) or by a slight abuse of notation:[\langle W, \varphi \rangle = \langle S_{\bf x} \otimes T_{\bf y}, \varphi ({\bf x} + {\bf y}) \rangle.]

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 [{\bb R}^{n}], then [\sigma^{-1} (K)] is a cylinder with base K and generator the `second bisector' [{\bf x} + {\bf y} = {\bf 0}] in [{\bb R}^{n} \times {\bb R}^{n}]. However, [\langle S \otimes T, \varphi \circ \sigma \rangle] is defined whenever the intersection between Supp [(S \otimes T) = (\hbox{Supp } S) \times (\hbox{Supp } T)] and [\sigma^{-1} (\hbox{Supp } \varphi)] is compact.

We may therefore define the convolution [S * T] of two distributions S and T on [{\bb R}^{n}] by[\langle S * T, \varphi \rangle = \langle S \otimes T, \varphi \circ \sigma \rangle = \langle S_{\bf x} \otimes T_{\bf y}, \varphi ({\bf x} + {\bf y})\rangle]whenever the following support condition is fulfilled:

`the set [\{({\bf x},{\bf y})|{\bf x} \in A, {\bf y} \in B, {\bf x} + {\bf y} \in K\}] is compact in [{\bb R}^{n} \times {\bb R}^{n}] for all K compact in [{\bb R}^{n}]'.

The latter condition is met, in particular, if S or T has compact support. The support of [S * T] is easily seen to be contained in the closure of the vector sum[A + B = \{{\bf x} + {\bf y}|{\bf x} \in A, {\bf y} \in B\}.]

Convolution by a fixed distribution S is a continuous operation for the topology on [{\scr D}\,']: it maps convergent sequences [(T_{j})] to convergent sequences [(S * T_{j})]. Convolution is commutative: [S * T = T * S].

The convolution of p distributions [T_{1}, \ldots, T_{p}] with supports [A_{1}, \ldots, A_{p}] can be defined by[\langle T_{1} * \ldots * T_{p}, \varphi \rangle = \langle (T_{1})_{{\bf x}_{1}} \otimes \ldots \otimes (T_{p})_{{\bf x}_{p}}, \varphi ({\bf x}_{1} + \ldots + {\bf x}_{p})\rangle]whenever the following generalized support condition:

`the set [\{({\bf x}_{1}, \ldots, {\bf x}_{p})|{\bf x}_{1} \in A_{1}, \ldots, {\bf x}_{p} \in A_{p}, {\bf x}_{1} + \ldots + {\bf x}_{p} \in K\}] is compact in [({\bb R}^{n})^{\,p}] for all K compact in [{\bb R}^{n}]'

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[link], pp. 436–437).

It follows from previous definitions that, for all distributions [T \in {\scr D}\,'], the following identities hold:

  • (i) [\delta * T = T]: [\delta] is the unit convolution;

  • (ii) [\delta_{({\bf a})} * T = \tau_{\bf a} T]: translation is a convolution with the corresponding translate of δ;

  • (iii) [(D^{{\bf p}} \delta) * T = D^{{\bf p}} T]: 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[link] 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 [T * \alpha] is an infinitely differentiable ordinary function. Since sequences [(\alpha_{\nu})] 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 [T * \alpha_{\nu}]. In topological jargon: [{\scr D}({\bb R}^{n})] is `everywhere dense' in [{\scr D}\,'({\bb R}^{n})]. A standard function in [{\scr D}] which is often used for such proofs is defined as follows: put[\eqalign{\theta (x) &= {1 \over A} \exp \left(- {1 \over 1-x^{2}}\right){\hbox to 10.5pt{}} \hbox{for } |x| \leq 1, \cr &= 0 \phantom{\exp \left(- {1 \over x^{2} - 1}\right)a}\quad \hbox{for } |x| \geq 1,}]with[A = \int\limits_{-1}^{+1} \exp \left(- {1 \over 1-x^{2}}\right) \,\hbox{d}x](so that θ is in [{\scr D}] and is normalized), and put[\eqalign{\theta_{\varepsilon} (x) &= {1 \over \varepsilon} \theta \left({x \over \varepsilon}\right){\hbox to 13.5pt{}}\hbox{ in dimension } 1,\cr \theta_{\varepsilon} ({\bf x}) &= \prod\limits_{j=1}^{n} \theta_{\varepsilon} (x_{j})\quad \hbox{in dimension } n.}]

Another related result, also proved by convolution, is the structure theorem: the restriction of a distribution [T \in {\scr D}\,'({\bb R}^{n})] to a bounded open set Ω in [{\bb R}^{n}] is a derivative of finite order of a continuous function.

Properties (i)[link] to (iv)[link] are the basis of the symbolic or operational calculus (see Carslaw & Jaeger, 1948[link]; Van der Pol & Bremmer, 1955[link]; Churchill, 1958[link]; Erdélyi, 1962[link]; Moore, 1971[link]) for solving integro-differential equations with constant coefficients by turning them into convolution equations, then using factorization methods for convolution algebras (Schwartz, 1965[link]).

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.








































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