Tensor products. Fubini's theorem

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. 27-28 | 1 | 2 |

Section 1.3.2.2.5. Tensor products. Fubini's theorem

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

1.3.2.2.5. Tensor products. Fubini's theorem

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Let $[f \in L^{1} ({\bb R}^{m})]$ , $[g \in L^{1} ({\bb R}^{n})]$ . Then the function $[f \otimes g: ({\bf x},{\bf y}) \;\longmapsto\; f({\bf x}) g({\bf y})]$ is called the tensor product of f and g, and belongs to $[L^{1} ({\bb R}^{m} \times {\bb R}^{n})]$ . The finite linear combinations of functions of the form $[f \otimes g]$ span a subspace of $[L^{1} ({\bb R}^{m} \times {\bb R}^{n})]$ called the tensor product of $[L^{1} ({\bb R}^{m})]$ and $[L^{1} ({\bb R}^{n})]$ and denoted $[L^{1} ({\bb R}^{m}) \otimes L^{1} ({\bb R}^{n})]$ .

The integration of a general function over $[{\bb R}^{m} \times {\bb R}^{n}]$ may be accomplished in two steps according to Fubini's theorem. Given $[F \in L^{1} ({\bb R}^{m} \times {\bb R}^{n})]$ , the functions $[\eqalign{F_{1} : {\bf x} &\;\longmapsto\; {\textstyle\int\limits_{{\bb R}^{n}}} F ({\bf x},{\bf y}) \;\hbox{d}^{n} {\bf y}\cr F_{2} : {\bf y} &\;\longmapsto\; {\textstyle\int\limits_{{\bb R}^{m}}} F ({\bf x},{\bf y}) \;\hbox{d}^{m} {\bf x}}]$ exist for almost all $[{\bf x} \in {\bb R}^{m}]$ and almost all $[{\bf y} \in {\bb R}^{n}]$ , respectively, are integrable, and $[\textstyle\int\limits_{{\bb R}^{m} \times {\bb R}^{n}} F ({\bf x},{\bf y}) \;\hbox{d}^{m} {\bf x} \;\hbox{d}^{n} {\bf y} = {\textstyle\int\limits_{{\bb R}^{m}}} F_{1} ({\bf x}) \;\hbox{d}^{m} {\bf x} = {\textstyle\int\limits_{{\bb R}^{n}}} F_{2} ({\bf y}) \;\hbox{d}^{n} {\bf y}.]$ Conversely, if any one of the integrals $[\displaylines{\quad (\hbox{i})\qquad {\textstyle\int\limits_{{\bb R}^{m} \times {\bb R}^{n}}} |F ({\bf x},{ \bf y})| \;\hbox{d}^{m} {\bf x} \;\hbox{d}^{n} {\bf y}\qquad \hfill\cr \quad (\hbox{ii})\qquad {\textstyle\int\limits_{{\bb R}^{m}}} \left({\textstyle\int\limits_{{\bb R}^{n}}} |F ({\bf x},{ \bf y})| \;\hbox{d}^{n} {\bf y}\right) \;\hbox{d}^{m} {\bf x}\hfill\cr \quad (\hbox{iii})\qquad {\textstyle\int\limits_{{\bb R}^{n}}} \left({\textstyle\int\limits_{{\bb R}^{m}}} |F ({\bf x},{ \bf y})| \;\hbox{d}^{m} {\bf x}\right) \;\hbox{d}^{n} {\bf y}\hfill}]$ is finite, then so are the other two, and the identity above holds. It is then (and only then) permissible to change the order of integrations.

Fubini's theorem is of fundamental importance in the study of tensor products and convolutions of distributions.

References

International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 27-28