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A General Perron Integral

Published online by Cambridge University Press:  20 November 2018

P. S. Bullen
P. S. Bullen*
Affiliation:
University of British Columbia
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In this paper integrals are considered from the point of view of inverting differential operators. In order to do this it is necessary to introduce integrals more general than the Lebesgue integral and these integrals turn out to have other interesting properties (6, 7, 12). The integral introduced here is defined in the setting of axiomatic potential theory (2, 4). By defining it as generally as possible it not only includes the James P2-integral but inverts many of the standard second-order differential operators.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 17 , 1965 , pp. 17 - 30
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Arsove, M. G., Functions representable as differences of subharmonic functions, Trans. Amer. Math. Soc, 75 (1953), 327–365.Google Scholar
2. Bauer, H., Axiomatische Behandlung des Dirichletschen Problems für elliptische und parabolische Differentialgleichungen, Math. Ann., 146 (1962), 1–59.Google Scholar
3. Brelot, M., Eléments de la théorie classique du potentiel (Paris, 1959).Google Scholar
4. Brelot, M., Lectures on potential theory (Bombay, 1960).Google Scholar
5. Brelot, M. and Choquet, G., Espaces et lignes de Green, Ann. Inst. Fourier, 3 (1952), 199–263.Google Scholar
6. James, R. D. and Gage, W. H., A generalised integral, Trans. Roy. Soc. Can., 3rd Ser., Sec. I l l, 40 (1946), 25–35.Google Scholar
7. James, R. D., A generalised integral II, Can. J. Math., 2 (1950), 297–306.Google Scholar
8. Rudin, W., Integral representation of continuous functions, Trans. Amer. Math. Soc, 70 (1951), 387–403.Google Scholar
9. Rudin, W., Uniqueness theory for Hermite series, Trans. Amer. Math. Soc, 70 (1951), 387–403.Google Scholar
10. Rudin, W., A theorem on subharmonic functions, Proc Amer. Math. Soc, 2 (1951), 209–212.Google Scholar
11. Rudin, W., Inversion of second order differential operators, Proc. Amer. Math. Soc, 3 (1952), 92–98.Google Scholar
12. Zygmund, A., Trigonometrical series, vol. 2 (Cambridge, 1959).Google Scholar