Integration of distributions in dimension 1

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, p. 32 | 1 | 2 |

Section 1.3.2.3.9.2. Integration of distributions in dimension 1

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

1.3.2.3.9.2. 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.

References

International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 32