International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 31-32   | 1 | 2 |

Section 1.3.2.3.9.1. Differentiation

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

1.3.2.3.9.1. 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{\semi}}]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 1.3.4.4.3.5[link] to calculate the Fourier transform of the indicator function of a molecular envelope.

References

Dirac, P. A. M. (1958). The Principles of Quantum Mechanics, 4th ed. Oxford: Clarendon Press.








































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