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Distributions defined as limits I. Distributions as derivatives; continuity

Published online by Cambridge University Press:  24 October 2008

J. R. Ravetz
Affiliation:
Durham University

Extract

The theory of distributions of L. Schwartz (3) provides a unified and rigorous foundation for special methods used in various branches of mathematics. Schwartz's treatment is on the most general level, and presupposes an understanding of modern abstract analysis. Several alternative approaches to distributions have been developed, all of them ‘elementary’ in one sense or another. We follow here the approach of Mikusiński (2) and Temple (4), in which distributions are defined as generalized limits of sequences of continuous functions. We find that, with this approach, it is possible to prove the basic theorem: every distribution is (locally) a derivative. The property of continuity of a distribution does not enter into the arguments establishing this result, but instead follows from it. Hence we are able to reduce the ‘regular sequence’ definition of a distribution to its simplest form. In a later paper we shall study convolution products of distributions, defined in the natural manner by regular sequences.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1957

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References

REFERENCES

(1)Bourbaki, N.Topologie générale, Chapitre IX (Paris, 1948).Google Scholar
(2)Mikusiński, J. G.Sur la méthode de généralisation de M. Laurent Schwartz et sur la con-vergence faible. Fundam. Math. 35 (1948), 235–9.CrossRefGoogle Scholar
(3)Schwartz, L.Théorie des distributions (Paris, 1950–1).Google Scholar
(4)Temple, G.The theory of generalized functions. Proc. Roy. Soc. A, 228 (1955), 175–90.Google Scholar