Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 28-29   | 1 | 2 |

Section Origins

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 Origins

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At the end of the 19th century, Heaviside proposed under the name of `operational calculus' a set of rules for solving a class of differential, partial differential and integral equations encountered in electrical engineering (today's `signal processing'). These rules worked remarkably well but were devoid of mathematical justification (see Whittaker, 1928[link]). In 1926, Dirac introduced his famous δ-function [see Dirac (1958)[link], pp. 58–61], which was found to be related to Heaviside's constructs. Other singular objects, together with procedures to handle them, had already appeared in several branches of analysis [Cauchy's `principal values'; Hadamard's `finite parts' (Hadamard, 1932[link], 1952[link]); Riesz's regularization methods for certain divergent integrals (Riesz, 1938[link], 1949[link])] as well as in the theories of Fourier series and integrals (see e.g. Bochner, 1932[link], 1959[link]). Their very definition often verged on violating the rigorous rules governing limiting processes in analysis, so that subsequent recourse to limiting processes could lead to erroneous results; ad hoc precautions thus had to be observed to avoid mistakes in handling these objects.

In 1945–1950, Laurent Schwartz proposed his theory of distributions (see Schwartz, 1966[link]), which provided a unified and definitive treatment of all these questions, with a striking combination of rigour and simplicity. Schwartz's treatment of Dirac's δ-function illustrates his approach in a most direct fashion. Dirac's original definition reads:[\displaylines{\quad (\hbox{i})\,\quad\delta ({\bf x}) = 0 \hbox{ for } {\bf x} \neq {\bf 0},\hfill\cr \quad (\hbox{ii})\quad {\textstyle\int_{{\bb R}^{n}}} \delta ({\bf x}) \,\hbox{d}^{n} {\bf x} = 1.\hfill}]These two conditions are irreconcilable with Lebesgue's theory of integration: by (i), δ vanishes almost everywhere, so that its integral in (ii) must be 0, not 1.

A better definition consists in specifying that[\displaylines{\quad (\hbox{iii})\quad {\textstyle\int_{{\bb R}^{n}}} \delta ({\bf x}) \varphi ({\bf x}) \,\hbox{d}^{n} {\bf x} = \varphi ({\bf 0})\hfill}]for any function ϕ sufficiently well behaved near [{\bf x} = {\bf 0}]. This is related to the problem of finding a unit for convolution (Section[link]). As will now be seen, this definition is still unsatisfactory. Let the sequence [(\,f_{\nu})] in [L^{1} ({\bb R}^{n})] be an approximate convolution unit, e.g.[f_{\nu} ({\bf x}) = \left({\nu \over 2\pi}\right)^{1/2} \exp (-{\textstyle{1 \over 2}} \nu^{2} \|{\bf x}\|^{2}).]Then for any well behaved function ϕ the integrals[{\textstyle\int\limits_{{\bb R}^{n}}} f_{\nu} ({\bf x}) \varphi ({\bf x}) \,\hbox{d}^{n} {\bf x}]exist, and the sequence of their numerical values tends to [\varphi ({\bf 0})]. It is tempting to combine this with (iii) to conclude that δ is the limit of the sequence [(\,f_{\nu})] as [\nu \rightarrow \infty]. However,[\lim f_{\nu} ({\bf x}) = 0 \quad \hbox{as } \nu \rightarrow \infty]almost everywhere in [{\bb R}^{n}] and the crux of the problem is that[\eqalign{\varphi ({\bf 0}) &= \lim\limits_{\nu \rightarrow \infty} {\textstyle\int\limits_{{\bb R}^{n}}} f_{\nu} ({\bf x}) \varphi ({\bf x}) \,\hbox{d}^{n} {\bf x} \cr &\neq {\textstyle\int\limits_{{\bb R}^{n}}} \left[\lim\limits_{\nu \rightarrow \infty} f_{v} ({\bf x}) \right] \varphi ({\bf x}) \,\hbox{d}^{n} {\bf x} = 0}]because the sequence [(\,f_{\nu})] does not satisfy the hypotheses of Lebesgue's dominated convergence theorem.

Schwartz's solution to this problem is deceptively simple: the regular behaviour one is trying to capture is an attribute not of the sequence of functions [(\,f_{\nu})], but of the sequence of continuous linear functionals[T_{\nu}: \varphi \,\longmapsto\, {\textstyle\int\limits_{{\bb R}^{n}}} f_{\nu} ({\bf x}) \varphi ({\bf x}) \,\hbox{d}^{n} {\bf x}]which has as a limit the continuous functional[T: \varphi \,\longmapsto\, \varphi ({\bf 0}).]It is the latter functional which constitutes the proper definition of δ. The previous paradoxes arose because one insisted on writing down the simple linear operation T in terms of an integral.

The essence of Schwartz's theory of distributions is thus that, rather than try to define and handle `generalized functions' via sequences such as [(\,f_{\nu})] [an approach adopted e.g. by Lighthill (1958)[link] and Erdélyi (1962)[link]], one should instead look at them as continuous linear functionals over spaces of well behaved functions.

There are many books on distribution theory and its applications. The reader may consult in particular Schwartz (1965[link], 1966[link]), Gel'fand & Shilov (1964)[link], Bremermann (1965)[link], Trèves (1967)[link], Challifour (1972)[link], Friedlander (1982)[link], and the relevant chapters of Hörmander (1963)[link] and Yosida (1965)[link]. Schwartz (1965)[link] is especially recommended as an introduction.


Bochner, S. (1932). Vorlesungen über Fouriersche Integrale. Leipzig: Akademische Verlagsgesellschaft.
Bochner, S. (1959). Lectures on Fourier Integrals. Translated from Bochner (1932) by M. Tenenbaum & H. Pollard. Princeton University Press.
Bremermann, H. (1965). Distributions, Complex Variables, and Fourier Transforms. Reading: Addison-Wesley.
Challifour, J. L. (1972). Generalized Functions and Fourier Analysis. Reading: Benjamin.
Dirac, P. A. M. (1958). The Principles of Quantum Mechanics, 4th ed. Oxford: Clarendon Press.
Erdélyi, A. (1962). Operational Calculus and Generalized Functions. New York: Holt, Rinehart & Winston.
Friedlander, F. G. (1982). Introduction to the Theory of Distributions. Cambridge University Press.
Gel'fand, I. M. & Shilov, G. E. (1964). Generalized Functions, Vol. I. New York, London: Academic Press.
Hadamard, J. (1932). Le Problème de Cauchy et les Equations aux Dérivées Partielles Linéaires Hyperboliques. Paris: Hermann.
Hadamard, J. (1952). Lectures on Cauchy's Problem in Linear Partial Differential Equations. New York: Dover Publications.
Hörmander, L. (1963). Linear Partial Differential Operators. Berlin: Springer-Verlag.
Lighthill, M. J. (1958). Introduction to Fourier Analysis and Generalized Functions. Cambridge University Press.
Riesz, M. (1938). L'intégrale de Riemann–Liouville et le problème de Cauchy pour l'équation des ondes. Bull. Soc. Math. Fr. 66, 153–170.
Riesz, M. (1949). L'intégrale de Riemann–Liouville et le problème de Cauchy. Acta Math. 81, 1–223.
Schwartz, L. (1965). Mathematics for the Physical Sciences. Paris: Hermann, and Reading: Addison-Wesley.
Schwartz, L. (1966). Théorie des Distributions. Paris: Hermann.
Trèves, F. (1967). Topological Vector Spaces, Distributions, and Kernels. New York, London: Academic Press.
Whittaker, E. T. (1928). Oliver Heaviside. Bull. Calcutta Math. Soc. 20, 199–220. [Reprinted in Moore (1971).]
Yosida, K. (1965). Functional Analysis. Berlin: Springer-Verlag.

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