Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T07:24:51.876Z Has data issue: false hasContentIssue false
  • English
  • Français

On generalized functions

Published online by Cambridge University Press:  14 July 2016

B. C. Rennie
Get access Rights & Permissions [Opens in a new window]

Abstract

There are described in the literature many spaces of what are variously described as generalized functions, distributions, or improper functions. This article introduces another. The new space is like that of M. J. Lighthill in containing the Fourier transform of every element and in having a particularly simple theory of trigonometric and Fourier series; also it is constructed in a somewhat similar way. The new space breaks away from the tradition of every element being, for some n, the nth derivative of an ordinary function, and, for example, the exponential function and its Fourier transform are in the space.

Keywords

DISTRIBUTIONSIMPROPER FUNCTIONSFOURIER SERIESFOURIER TRANSFORMSGENERALIZED FUNCTIONS
Type
Part 3 — Mathematics
Information
Journal of Applied Probability , Volume 19 , Issue A , 1982 , pp. 139 - 156
Copyright
Copyright © 1982 Applied Probability Trust 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Burkill, J. C. (1957) An integral for distributions. Proc. Camb. Phil. Soc. 53, 821–824.CrossRefGoogle Scholar
[2] Dirac, P. A. M. (1947) The Principles of Quantum Mechanics , 3rd edn. Clarendon Press, Oxford.Google Scholar
[3] Eddington, A. S. (1936) Relativity Theory of Protons and Electrons. Cambridge University Press, London.Google Scholar
[4] Hardy, G. H. (1933) A theorem concerning Fourier transforms. J. London Math. Soc. 8, 227–231.CrossRefGoogle Scholar
[5] Lighthill, M. J. (1958) Fourier Analysis and Generalised Functions. Cambridge University Press. London.Google Scholar
[6] Schwartz, L. (1957) Théorie des distributions , I. Actualités Scientifiques et Industrielles , 1091, 1245.Google Scholar
[7] Temple, G. (1953) Theories and applications of generalized functions. J. London Math. Soc. 28, 134–148.CrossRefGoogle Scholar