Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 32   | 1 | 2 |

Section Integration of distributions in dimension 1

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 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.

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