Convolution of distributions

Bricogne, G.

doi:10.1107/97809553602060000551

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

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International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 33-34 | 1 | 2 |

Section 1.3.2.3.9.7. Convolution of distributions

G. Bricogne^a

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

1.3.2.3.9.7. 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, 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 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) 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

Erdélyi, A. (1962). Operational calculus and generalized functions. New York: Holt, Rinehart & Winston.Google Scholar

Moore, D. H. (1971). Heaviside operational calculus. An elementary foundation. New York: American Elsevier.Google Scholar

Schwartz, L. (1965). Mathematics for the physical sciences. Paris: Hermann, and Reading: Addison-Wesley.Google Scholar

Van der Pol, B. & Bremmer, H. (1955). Operational calculus, 2nd ed. Cambridge University Press.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 33-34