Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-20T06:11:00.779Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Applications of a Theorem of Marcinkiewicz

Published online by Cambridge University Press:  20 November 2018

P. S. Bullen  and
R. Vyborny
P. S. Bullen
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, Canada
R. Vyborny
Affiliation:
Department of Mathematics, University of Queensland, St. Lucia, Queensland, Australia
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.

A classical theorem of Marcinkiewicz states that a function is Perron integrable iff it has one continuous major and one continuous minor function. Using an elaboration of this remarkable theorem three applications are made; to obtain a new proof of a recent characterization of the Perron integral, to proofs of some theorems on interchange of limits and integration and to extend classical existence theorems for ordinary differential equations.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 34 , Issue 2 , 01 June 1991 , pp. 165 - 174
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Bullen, P. S. Non-absolute integrals: a survey, Real Analysis Exchange, 5 (1980), 195–259.Google Scholar
2. Burkill, J. C. The Cesaro-Perron integral, Proc. London Math Soc. (2) 34 (1932), 314–322.Google Scholar
3. Coddington, E. A., Levinson, N. Theory of Ordinary Differential Equations. McGraw-Hill, New York. Toronto, London 1955.Google Scholar
4. Denjoy, A. Leçons sur le Calcul de Coefficients d'une Série Trigonometrique, Vol. IV, Paris, 1949.Google Scholar
5. Henstock, R. Theory of Integration, London, 1963.Google Scholar
6. Kurzweil, J. Nichtabsolut konvergente Intégrale, Leipzig, 1980.Google Scholar
7. Lee, P. Y. and Chew, T. S. On convergence theorems for non-absolute integrals, Bull. Austral. Math. Soc, 34(1986), 133–140.Google Scholar
8. Lee, P. Y and Lu, S. Notes on Classical Integration Theory (viii), Res. Rep. No 327, Dept. Math. Univ. Singapore 1988.Google Scholar
9. Mawhin, J. Introduction a l'analyse, Louvain in-la-Neuve, Cobay, 1984.Google Scholar
10. McShane, E. J. Unified Integration, New York, 1983.Google Scholar
11. Pfeffer, W. F. The Riemann-Stieltjes approach to integration, Tech. Rep. Nat. Res. Inst. Math. Sci., Pretoria, 1980.Google Scholar
12. Saks, S. The Theory of the Integral, 2nd Ed. revised, New York, 1937.Google Scholar
13. Sarkhel, D. N. A criterion for Perron integrability, Proc. Amer. Math. Soc, 70 (1978), 109–112.Google Scholar
14. Schurle, A. W. A function is Perron integrable if it has locally small Riemann sums, J. Austral. Math. Soc. (Ser. A), 41 (1986), 224–232.Google Scholar
15. Schurle, A. W. A new property equivalent to Lebesgue integrability, Proc. Amer. Math. Soc, 96 (1986), 103–106.Google Scholar
16. Tolstov, G. P. Sur l'intégrale de Perron, Mat. Sb., 5 (1939) 647–649.Google Scholar
17. Vyborny, R. A remark on Perron's method in the proof of the Peano's Theorem. Acta. Math. Sci. 5 (1985), 349–352.Google Scholar