Differentiation

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. 31-32 | 1 | 2 |

Section 1.3.2.3.9.1. Differentiation

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

1.3.2.3.9.1. 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 $[\eqalign{&{\textstyle\int\limits_{{\bb R}^{n}}} \partial_{i}\; f(x_{\rm l}, \ldots, x_{i}, \ldots, x_{n}) \varphi (x_{\rm l}, \ldots, x_{i}, \ldots, x_{n}) \;\hbox{d}x_{i} \cr &\quad = (\;f\varphi)(x_{\rm l}, \ldots, + \infty, \ldots, x_{n}) - (\;f\varphi)(x_{\rm l}, \ldots, - \infty, \ldots, x_{n}) \cr &\qquad - {\textstyle\int\limits_{{\bb R}^{n}}} f(x_{\rm l}, \ldots, x_{i}, \ldots, x_{n}) \partial_{i} \varphi (x_{\rm l}, \ldots, x_{i}, \ldots, x_{n}) \;\hbox{d}x_{i}\hbox{;}}]$ 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)].

Let f be infinitely differentiable for $[x \;\lt\; 0]$ and $[x \gt 0]$ but have discontinuous derivatives $[f^{(m)}]$ at [ $[\;f^{(0)}]$ 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 to calculate the Fourier transform of the indicator function of a molecular envelope.