Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 33-34   | 1 | 2 |

Section Tensor product 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 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.

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