Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

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

Section 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 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 [\varphi \in {\scr D}] by `transposing' it in the duality product [\langle T, \varphi \rangle]; this procedure will map T to a new distribution provided the original operation maps [{\scr D}] continuously into itself. Differentiation

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  • (a) Definition and elementary properties

    If T is a distribution on [{\bb R}^{n}], its partial derivative [\partial_{i} T] with respect to [x_{i}] is defined by [\langle \partial_{i} T, \varphi \rangle = - \langle T, \partial_{i} \varphi \rangle]

    for all [\varphi \in {\scr D}]. This does define a distribution, because the partial differentiations [\varphi \;\longmapsto\; \partial_{i} \varphi] are continuous for the topology of [{\scr D}].

    Suppose that [T = T_{f}] with f a locally integrable function such that [\partial_{i}\; f] exists and is almost everywhere continuous. Then integration by parts along the [x_{i}] axis gives [\eqalign{&{\textstyle\int\limits_{{\bb R}^{n}}} \partial_{i}\; f(x_{\rm l}, \ldots, x_{i}, \ldots, x_{n}) \varphi (x_{\rm l}, \ldots, x_{i}, \ldots, x_{n}) \;\hbox{d}x_{i} \cr &\quad = (\;f\varphi)(x_{\rm l}, \ldots, + \infty, \ldots, x_{n}) - (\;f\varphi)(x_{\rm l}, \ldots, - \infty, \ldots, x_{n}) \cr &\qquad - {\textstyle\int\limits_{{\bb R}^{n}}} f(x_{\rm l}, \ldots, x_{i}, \ldots, x_{n}) \partial_{i} \varphi (x_{\rm l}, \ldots, x_{i}, \ldots, x_{n}) \;\hbox{d}x_{i}\hbox{;}}] the integrated term vanishes, since ϕ has compact support, showing that [\partial_{i} T_{f} = T_{\partial_{i}\; f}].

    The test functions [\varphi \in {\scr D}] are infinitely differentiable. Therefore, transpositions like that used to define [\partial_{i} T] may be repeated, so that any distribution is infinitely differentiable. For instance, [\displaylines{\langle \partial_{ij}^{2} T, \varphi \rangle = - \langle \partial_{j} T, \partial_{i} \varphi \rangle = \langle T, \partial_{ij}^{2} \varphi \rangle, \cr \langle D^{\bf p} T, \varphi \rangle = (-1)^{|{\bf p}|} \langle T, D^{\bf p} \varphi \rangle, \cr \langle \Delta T, \varphi \rangle = \langle T, \Delta \varphi \rangle,}] where Δ is the Laplacian operator. The derivatives of Dirac's δ distribution are [\langle D^{\bf p} \delta, \varphi \rangle = (-1)^{|{\bf p}|} \langle \delta, D^{\bf p} \varphi \rangle = (-1)^{|{\bf p}|} D^{\bf p} \varphi ({\bf 0}).]

    It is remarkable that differentiation is a continuous operation for the topology on [{\scr D}\,']: if a sequence [(T_{j})] of distributions converges to distribution T, then the sequence [(D^{\bf p} T_{j})] of derivatives converges to [D^{\bf p} T] for any multi-index p, since as [j \rightarrow \infty] [\langle D^{\bf p} T_{j}, \varphi \rangle = (-1)^{|{\bf p}|} \langle T_{j}, D^{\bf p} \varphi \rangle \rightarrow (-1)^{|{\bf p}|} \langle T, D^{\bf p} \varphi \rangle = \langle D^{\bf p} T, \varphi \rangle.] 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 [\langle ,\rangle] as under the integral sign with ordinary functions. Let the function [\varphi = \varphi ({\bf x}, \lambda)] depend on a parameter [\lambda \in \Lambda] and a vector [{\bf x} \in {\bb R}^{n}] in such a way that all functions [\varphi_{\lambda}: {\bf x} \;\longmapsto\; \varphi ({\bf x}, \lambda)] be in [{\scr D}({\bb R}^{n})] for all [\lambda \in \Lambda]. Let [T \in {\scr D}^{\prime}({\bb R}^{n})] be a distribution, let [I(\lambda) = \langle T, \varphi_{\lambda}\rangle] and let [\lambda_{0} \in \Lambda] be given parameter value. Suppose that, as λ runs through a small enough neighbourhood of [\lambda_{0}],

    • (i) all the [\varphi_{\lambda}] have their supports in a fixed compact subset K of [{\bb R}^{n}];

    • (ii) all the derivatives [D^{\bf p} \varphi_{\lambda}] have a partial derivative with respect to λ which is continuous with respect to x and λ.

    Under these hypotheses, [I(\lambda)] is differentiable (in the usual sense) with respect to λ near [\lambda_{0}], and its derivative may be obtained by `differentiation under the [\langle ,\rangle] sign': [{\hbox{d}I \over \hbox{d}\lambda} = \langle T, \partial_{\lambda} \varphi_{\lambda}\rangle.]

  • (c) Effect of discontinuities

    When a function f or its derivatives are no longer continuous, the derivatives [D^{\bf p} T_{f}] of the associated distribution [T_{f}] may no longer coincide with the distributions associated to the functions [D^{\bf p} f].

    In dimension 1, the simplest example is Heaviside's unit step function [Y\; [Y(x) = 0 \hbox{ for } x \;\lt\; 0, Y(x) = 1 \hbox{ for } x \geq 0]]: [\langle (T_{Y})', \varphi \rangle = - \langle (T_{Y}), \varphi'\rangle = - {\textstyle\int\limits_{0}^{+ \infty}} \varphi' (x) \;\hbox{d}x = \varphi (0) = \langle \delta, \varphi \rangle.] Hence [(T_{Y})' = \delta], a result long used `heuristically' by electrical engineers [see also Dirac (1958)[link]].

    Let f be infinitely differentiable for [x \;\lt\; 0] and [x \gt 0] but have discontinuous derivatives [f^{(m)}] at [x = 0] [[\;f^{(0)}] being f itself] with jumps [\sigma_{m} = f^{(m)} (0 +) - f^{(m)} (0 -)]. Consider the functions: [\eqalign{g_{0} &= f - \sigma_{0} Y \cr g_{1} &= g'_{0} - \sigma_{1} Y \cr---&-------\cr g_{k} &= g'_{k - 1} - \sigma_{k} Y.}] The [g_{k}] are continuous, their derivatives [g'_{k}] are continuous almost everywhere [which implies that [(T_{g_{k}})' = T_{g'_{k}}] and [g'_{k} = f^{(k + 1)}] almost everywhere]. This yields immediately: [\eqalign{(T_{f})' &= T_{f'} + \sigma_{0} \delta \cr (T_{f})'' &=T_{f''} + \sigma_{0} \delta' + \sigma_{\rm 1} \delta \cr----&--------------\cr (T_{f})^{(m)} &= T_{f^{(m)}} + \sigma_{0} \delta^{(m - 1)} + \ldots + \sigma_{m - 1} \delta.\cr----&--------------\cr}] Thus the `distributional derivatives' [(T_{f})^{(m)}] differ from the usual functional derivatives [T_{f^{(m)}}] 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 [\sigma_{0}] and [\sigma_{\nu}] denote the discontinuities of f and its normal derivative [\partial_{\nu} \varphi] across S (both [\sigma_{0}] and [\sigma_{\nu}] are functions of position on S), and let [\delta_{(S)}] and [\partial_{\nu} \delta_{(S)}] be defined by [\eqalign{\langle \delta_{(S)}, \varphi \rangle &= {\textstyle\int\limits_{S}} \varphi \;\hbox{d}^{n - 1} S \cr \langle \partial_{\nu} \delta_{(S)}, \varphi \rangle &= - {\textstyle\int\limits_{S}} \partial_{\nu} \varphi \;\hbox{d}^{n - 1} S.}] Integration by parts shows that [\partial_{i} T_{f} = T_{\partial_{i}\; f} + \sigma_{0} \cos \theta_{i} \delta_{(S)},] where [\theta_{i}] is the angle between the [x_{i}] axis and the normal to S along which the jump [\sigma_{0}] occurs, and that the Laplacian of [T_{f}] is given by [\Delta (T_{f}) = T_{\Delta f} + \sigma_{\nu} \delta_{(S)} + \partial_{\nu} [\sigma_{0} \delta_{(S)}].] The latter result is a statement of Green's theorem in terms of distributions. It will be used in Section[link] to calculate the Fourier transform of the indicator function of a molecular envelope. 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 [T' = S].

For all [\chi \in {\scr D}] such that [\chi = \psi'] with [\psi \in {\scr D}], we must have [\langle T, \chi \rangle = - \langle S, \psi \rangle .] This condition defines T in a `hyperplane' [{\scr H}] of [{\scr D}], whose equation [\langle 1, \chi \rangle \equiv \langle 1, \psi' \rangle = 0] reflects the fact that ψ has compact support.

To specify T in the whole of [{\scr D}], it suffices to specify the value of [\langle T, \varphi_{0} \rangle] where [\varphi_{0} \in {\scr D}] is such that [\langle 1, \varphi_{0} \rangle = 1]: then any [\varphi \in {\scr D}] may be written uniquely as [\varphi = \lambda \varphi_{0} + \psi'] with [\lambda = \langle 1, \varphi \rangle, \qquad \chi = \varphi - \lambda \varphi_{0}, \qquad \psi (x) = {\textstyle\int\limits_{0}^{x}} \chi (t) \;\hbox{d}t,] and T is defined by [\langle T, \varphi \rangle = \lambda \langle T, \varphi_{0} \rangle - \langle S, \psi \rangle.] The freedom in the choice of [\varphi_{0}] means that T is defined up to an additive constant. Multiplication of distributions by functions

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The product [\alpha T] of a distribution T on [{\bb R}^{n}] by a function α over [{\bb R}^{n}] will be defined by transposition: [\langle \alpha T, \varphi \rangle = \langle T, \alpha \varphi \rangle \quad \hbox{for all } \varphi \in {\scr D}.] In order that [\alpha T] be a distribution, the mapping [\varphi \;\longmapsto\; \alpha \varphi] must send [{\scr D}({\bb R}^{n})] 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[link]) 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 [\alpha \delta = \alpha ({\bf 0}) \delta] 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[link],[link]). More generally, [D^{{\bf p}}\delta] is a distribution of order [|{\bf p}|], and the following formula holds for all [\alpha \in {\scr D}^{(m)}] with [m = |{\bf p}|]: [\alpha (D^{{\bf p}}\delta) = {\displaystyle\sum\limits_{{\bf q} \leq {\bf p}}} (-1)^{|{\bf p}-{\bf q}|} \pmatrix{{\bf p}\cr {\bf q}\cr} (D^{{\bf p}-{\bf q}} \alpha) ({\bf 0}) D^{\bf q}\delta.]

The derivative of a product is easily shown to be [\partial_{i}(\alpha T) = (\partial_{i}\alpha) T + \alpha (\partial_{i}T)] and generally for any multi-index p [D^{\bf p}(\alpha T) = {\displaystyle\sum\limits_{{\bf q}\leq {\bf p}}} \pmatrix{{\bf p}\cr {\bf q}\cr} (D^{{\bf p}-{\bf q}} \alpha) ({\bf 0}) D^{{\bf q}}T.] Division of distributions by functions

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Given a distribution S on [{\bb R}^{n}] and an infinitely differentiable multiplier function α, the division problem consists in finding a distribution T such that [\alpha T = S].

If α never vanishes, [T = S/\alpha] is the unique answer. If [n = 1], and if α has only isolated zeros of finite order, it can be reduced to a collection of cases where the multiplier is [x^{m}], for which the general solution can be shown to be of the form [T = U + {\textstyle\sum\limits_{i=0}^{m-1}} c_{i}\delta^{(i)},] where U is a particular solution of the division problem [x^{m} U = S] and the [c_{i}] are arbitrary constants.

In dimension [n \gt 1], 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)[link]]. Transformation of coordinates

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Let σ be a smooth non-singular change of variables in [{\bb R}^{n}], i.e. an infinitely differentiable mapping from an open subset Ω of [{\bb R}^{n}] to Ω′ in [{\bb R}^{n}], whose Jacobian [J(\sigma) = \det \left[{\partial \sigma ({\bf x}) \over \partial {\bf x}}\right]] vanishes nowhere in Ω. By the implicit function theorem, the inverse mapping [\sigma^{-1}] from Ω′ to Ω is well defined.

If f is a locally summable function on Ω, then the function [\sigma^{\#} f] defined by [(\sigma^{\#} f)({\bf x}) = f[\sigma^{-1}({\bf x})]] is a locally summable function on Ω′, and for any [\varphi \in {\scr D}(\Omega')] we may write: [\eqalign{{\textstyle\int\limits_{\Omega'}} (\sigma^{\#} f) ({\bf x}) \varphi ({\bf x}) \;\hbox{d}^{n} {\bf x} &= {\textstyle\int\limits_{\Omega'}} f[\sigma^{-1} ({\bf x})] \varphi ({\bf x}) \;\hbox{d}^{n} {\bf x} \cr &= {\textstyle\int\limits_{\Omega'}} f({\bf y}) \varphi [\sigma ({\bf y})]|J(\sigma)| \;\hbox{d}^{n} {\bf y} \quad \hbox{by } {\bf x} = \sigma ({\bf y}).}] In terms of the associated distributions [\langle T_{\sigma^{\#} f}, \varphi \rangle = \langle T_{f}, |J(\sigma)|(\sigma^{-1})^{\#} \varphi \rangle.]

This operation can be extended to an arbitrary distribution T by defining its image [\sigma^{\#} T] under coordinate transformation σ through [\langle \sigma^{\#} T, \varphi \rangle = \langle T, |J(\sigma)|(\sigma^{-1})^{\#} \varphi \rangle,] which is well defined provided that σ is proper, i.e. that [\sigma^{-1}(K)] is compact whenever K is compact.

For instance, if [\sigma: {\bf x} \;\longmapsto\; {\bf x} + {\bf a}] is a translation by a vector a in [{\bb R}^{n}], then [|J(\sigma)| = 1]; [\sigma^{\#}] is denoted by [\tau_{\bf a}], and the translate [\tau_{\bf a} T] of a distribution T is defined by [\langle \tau_{\bf a} T, \varphi \rangle = \langle T, \tau_{-{\bf a}} \varphi \rangle.]

Let [A: {\bf x} \;\longmapsto\; {\bf Ax}] be a linear transformation defined by a non-singular matrix A. Then [J(A) = \det {\bf A}], and [\langle A^{\#} T, \varphi \rangle = |\det {\bf A}| \langle T, (A^{-1})^{\#} \varphi \rangle.] This formula will be shown later (Sections[link],[link]) to be the basis for the definition of the reciprocal lattice.

In particular, if [{\bf A} = -{\bf I}], where I is the identity matrix, A is an inversion through a centre of symmetry at the origin, and denoting [A^{\#} \varphi] by [\breve{\varphi}] we have: [\langle \breve{T}, \varphi \rangle = \langle T, \breve{\varphi} \rangle.] T is called an even distribution if [\breve{T} = T], an odd distribution if [\breve{T} = -T].

If [{\bf A} = \lambda {\bf I}] with [\lambda \gt 0], A is called a dilation and [\langle A^{\#} T, \varphi \rangle = \lambda^{n} \langle T, (A^{-1})^{\#} \varphi \rangle.] Writing symbolically δ as [\delta ({\bf x})] and [A^{\#} \delta] as [\delta ({\bf x}/\lambda)], we have: [\delta ({\bf x}/\lambda) = \lambda^{n} \delta ({\bf x}).] If [n = 1] and f is a function with isolated simple zeros [x_{j}], then in the same symbolic notation [\delta [\;f(x)] = \sum\limits_{j} {1 \over |\;f'(x_{j})|} \delta (x_{j}),] where each [\lambda_{j} = 1/|\;f'(x_{j})|] is analogous to a `Lorentz factor' at zero [x_{j}]. Tensor product of distributions

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The purpose of this construction is to extend Fubini's theorem to distributions. Following Section[link], we may define the tensor product [L_{\rm loc}^{1} ({\bb R}^{m}) \otimes L_{\rm loc}^{1} ({\bb R}^{n})] as the vector space of finite linear combinations of functions of the form [f \otimes g: ({\bf x},{ \bf y}) \;\longmapsto\; f({\bf x})g({\bf y}),] where [{\bf x} \in {\bb R}^{m},{\bf y} \in {\bb R}^{n}, f \in L_{\rm loc}^{1} ({\bb R}^{m})] and [g \in L_{\rm loc}^{1} ({\bb R}^{n})].

Let [S_{\bf x}] and [T_{\bf y}] denote the distributions associated to f and g, respectively, the subscripts x and y acting as mnemonics for [{\bb R}^{m}] and [{\bb R}^{n}]. It follows from Fubini's theorem (Section[link]) that [f \otimes g \in L_{\rm loc}^{1} ({\bb R}^{m} \times {\bb R}^{n})], and hence defines a distribution over [{\bb R}^{m} \times {\bb R}^{n}]; the rearrangement of integral signs gives [\langle S_{\bf x} \otimes T_{\bf y}, \varphi_{{\bf x}, \,{\bf y}} \rangle = \langle S_{\bf x}, \langle T_{\bf y}, \varphi_{{\bf x}, \,{\bf y}} \rangle\rangle = \langle T_{\bf y}, \langle S_{\bf x}, \varphi_{{\bf x}, \, {\bf y}} \rangle\rangle] for all [\varphi_{{\bf x}, \,{\bf y}} \in {\scr D}({\bb R}^{m} \times {\bb R}^{n})]. In particular, if [\varphi ({\bf x},{ \bf y}) = u({\bf x}) v({\bf y})] with [u \in {\scr D}({\bb R}^{m}),v \in {\scr D}({\bb R}^{n})], then [\langle S \otimes T, u \otimes v \rangle = \langle S, u \rangle \langle T, v \rangle.]

This construction can be extended to general distributions [S \in {\scr D}\,'({\bb R}^{m})] and [T \in {\scr D}\,'({\bb R}^{n})]. Given any test function [\varphi \in {\scr D}({\bb R}^{m} \times {\bb R}^{n})], let [\varphi_{\bf x}] denote the map [{\bf y} \;\longmapsto\; \varphi ({\bf x}, {\bf y})]; let [\varphi_{\bf y}] denote the map [{\bf x} \;\longmapsto\; \varphi ({\bf x},{\bf y})]; and define the two functions [\theta ({\bf x}) = \langle T, \varphi_{\bf x} \rangle] and [\omega ({\bf y}) = \langle S, \varphi_{\bf y} \rangle]. Then, by the lemma on differentiation under the [\langle,\rangle] sign of Section[link], [\theta \in {\scr D}({\bb R}^{m}),\omega \in {\scr D}({\bb R}^{n})], and there exists a unique distribution [S \otimes T] such that [\langle S \otimes T, \varphi \rangle = \langle S, \theta \rangle = \langle T, \omega \rangle.] [S \otimes T] 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: [\langle S_{\bf x} \otimes T_{\bf y}, \varphi_{{\bf x}, \, {\bf y}} \rangle = \langle S_{\bf x}, \langle T_{\bf y}, \varphi_{{\bf x}, \, {\bf y}} \rangle\rangle = \langle T_{\bf y}, \langle S_{\bf x}, \varphi_{{\bf x}, \, {\bf y}} \rangle\rangle.]

The tensor product of distributions is associative: [(R \otimes S) \otimes T = R \otimes (S \otimes T).] Derivatives may be calculated by [D_{\bf x}^{\bf p} D_{\bf y}^{\bf q} (S_{\bf x} \otimes T_{\bf y}) = (D_{\bf x}^{\bf p} S_{\bf x}) \otimes (D_{\bf y}^{\bf q} T_{\bf y}).] The support of a tensor product is the Cartesian product of the supports of the two factors. 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[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]).


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