Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-20T03:21:29.674Z Has data issue: false hasContentIssue false
  • English
  • Français

The Exceptional Sets in the Definition of the Pn-Integral

Published online by Cambridge University Press:  20 November 2018

G. E. Cross
G. E. Cross*
Affiliation:
University of Waterloo, Waterloo, Ontario
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.

It has been recently observed by S. N. Mukhopadhyay that the various definitions of the Pn-integral are not complete unless it is shown that the exceptional scattered set allowed in the definition is not important. Utilizing the fact that on the real line a scattered set is countable, and adapting known methods for coping with exceptional countable sets, it is proved that the definitions of the Pn-integral are complete. It is then clear that the concept of scattered set is not essential to the definition of the Pn-integral.

Keywords

26A3926A51
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 25 , Issue 4 , 01 December 1982 , pp. 385 - 391
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Bosanquet, L. S., A property of Cesàro-Perron integrals, Proc. Edinburgh Math. Soc. (2) 6 (1940), 160-165.Google Scholar
2. Bullen, P. S., A Criterion for n-Convexity, Pacific J. Math. 36 (1971), 81-89.Google Scholar
3. Burkill, J. C., Integrals and Trigonometric Series, Proc. London Math. Soc. (3) 1 (1951), 46-57.Google Scholar
4. Cross, G. E., The Pn-integral, Canad. Math. Bull. 18 (1975), 493-497.Google Scholar
5. Cross, G. E., The Representation of (C, k) Summable Series in Fourier Form, Canad. Math. Bull. 21 (1978), 149-158.Google Scholar
6. Grimshaw, M. E., Thé Cauchy property of the generalized Perron integrals, Proc. Cambridge Phil. Soc. 30 (1934), 15-18.Google Scholar
7. James, R. D., A Generalized Integral II, Can. J. Math. 2 (1950), 297-306.Google Scholar
8. James, R. D., Generalized nth Primitives, Trans. Amer. Math. Soc. 76 (1954), 149-176.Google Scholar
9. James, R. D., Summable Trigonometric Series, Pacific J. Math. 6 (1956), 99-110.Google Scholar
10. James, R. D. and Gage, W. H., A Generalized Integral, Trans. Roy. Soc. Can. 40 (1946), 25-35.Google Scholar
11. Mukhopadhyay, S. N., On the Regularity of the Pn-integral, Pacific J. Math. 55 (1974), 233-247.Google Scholar
12. Semadeni, Z., Banach Spaces of Continuous Functions, Warsaw, 1971.Google Scholar
13. Taylor, S. J., An Integral of Perron's Type, Quart. J. Math. Oxford (2), 6 (1955), 255-274.Google Scholar