Hostname: page-component-7479d7b7d-c9gpj Total loading time: 0 Render date: 2024-07-13T16:40:26.873Z Has data issue: false hasContentIssue false
  • English
  • Français

The interrelations among various spaces of distributions

Published online by Cambridge University Press:  17 April 2009

S. Jeyamma
S. Jeyamma
Affiliation:
Madurai University, Madurai – 2, Tamil Nadu, India.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we discuss the interrelations among various spaces of distributions and show that none of them can be linearly and differentiably homeomorphic to the space of Mikusiński operators. It is also shown that the distributions of Mikusiński-Sikorski can also be defined by the method described by Temple as the completion of the space of continuous functions after introducing a weaker notion of convergence in this space.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 4 , Issue 2 , April 1971 , pp. 263 - 271
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Foias, C., “Approximations des opérateurs de J. Mikusiński par des fonctions continues”, Studia Math. 21 (1961/1962), 73–74.CrossRefGoogle Scholar
[2]Jeyamma, S. and Venkataraman, M., “On continuously differentiable spaces”, Publ. Math. Debrecen 18 (1971), (to appear).Google Scholar
[3]Lighthill, M.J., Introduction to Fourier analysis and generalized functions (Cambridge University Press, Cambridge, 1958).CrossRefGoogle Scholar
[4]Liverman, T.P.G., Generalized functions and direct operational methods (Prentice-Hall, Englewood Cliffs, New Jersey, 1964).Google Scholar
[5]Mikusiński, Jan, Operational calculus (International Series of Monographs on Pure and Applied Mathematics, 8. Pergamon Press, New York, London, Paris, Los Angeles; Pánstwowe Wydawnictwo Naukowe, Warsaw; 1959).Google Scholar
[6]-Mikusiński, Jan G., “Sur la méthode de généralisation de Laurent Schwartz et sur la convergence faible”, Fund. Math. 35 (1948), 235–239.CrossRefGoogle Scholar
[7]Mikusiński, J. and Sikorski, R., “The elementary theory of distrib distributions. I”, Rozprawy Mat. 12 (1957), 54 pages. “The elementary theory of distributions. II”, Rozprawy Mat. 25 (1961), 47 pages.Google Scholar
[8]Schwartz, Laurent, Théorie des distributions, Tomes I, II (Actualités Sci. Ind., nos. 1091, 1122 = Publ. Inst. Math. Univ. Strasborg 9, 10. Hermann, Paris, 1950, 1951).Google Scholar
[9]Temple, G., “Theories and applications of generalized functions”, J. London Math. Soc. 28 (1953), 134–148.CrossRefGoogle Scholar
[10]Zieleźny, Z., “On infinite derivatives of continuous functions”, Studia Math. 24 (1964), 311–351.CrossRefGoogle Scholar