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A General Perron Integral, II

Published online by Cambridge University Press:  20 November 2018

P. S. Bullen
P. S. Bullen*
Affiliation:
University of British Columbia
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This paper continues work begun in a previous paper of the same title (7), which will be called I; results from I will be referred to as Theorem 1.4, Axiom 1.1 etc. The notation used in the present paper will, except where noted, be that of I, to which reference should be made for further details.

In § 2, certain ideas presented in I are modified to give a neater and more general theory and then some new results of this theory are added. The remaining two sections develop some of the examples mentioned in I, § 5.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 457 - 473
Copyright
Copyright © Canadian Mathematical Society 1967

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. Bauer, H., Weiterführung einer axiomatischen Potentialtheorie ohne Kern (Existenz von Potentialen), Z. Wahrscheinlichkeitstheorie, 1 (1963), 197–229.Google Scholar
4. Bauer, H., Propriétés fines des fonctions surharmonique s dans une théorie axiomatique du potential, Colloq. Intern, du C.N.R.S. (Théorie du Potentiel), 1964.Google Scholar
5. Bonsall, F. F., On generalised subharmonic functions, Proc. Cambridge Philos. Soc., 46 (1950), 387–395.Google Scholar
6. Brelot, M. and Choquet, G., Espaces et lignes de Green, Ann. Inst. Fourier, Grenoble, 8 (1951), 199–263.Google Scholar
7. Bullen, P. S., A general Perron integral, Can. J. Math., 17 (1965), 17–30.Google Scholar
8. Burkill, J. C., The Cesàro-Perron scale of integration, Proc. London Math. Soc., 89, 2 (1935), 541–552.Google Scholar
9. Denjoy, A., Leçons sur le calcul des coefficients d'une série trigonométrique (Paris, 1941).Google Scholar
10. Doob, J., A probability approach to the heat equations, Trans. Amer. Math. Soc., 80 (1955), 216–280.Google Scholar
11. Dynkin, E. B., Markov processes (New York, 1965).Google Scholar
12. Gilbarg, D. and Serrin, J., On isolated singularities of second order elliptic differential equations, J. Analyse Math., 4 (1954-56), 309–340.Google Scholar
13. Hervé, R.-M., Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier, Grenoble, 12 (1962), 415–571.Google Scholar
14. Ito, S., On existence of Green function and positive superharmonic functions for linear elliptic operators of second order, J. Math. Soc. Japan, 16 (1964), 299–306.Google Scholar
15. James, R. D., A generalised integral, II, Can. J. Math., 2 (1950), 297–306.Google Scholar
16. James, R. D. and Gage, W. H., A generalised integral, Trans. Roy. Soc. Canada, 3rd Ser., Sec. III, 40 (1946), 25–35.Google Scholar
17. Meyer, P.-A., Brelot's axiomatic theory of the Dirichlet problem and Hunt's theory, Ann. Inst. Fourier, Grenoble, 18 (1963), 357–372.Google Scholar
18. Miranda, C., Equazione aile derivate parziali di tipo ellito (Berlin, 1955).Google Scholar
19. Rudin, W., Integral representation of continuous functions, Trans. Amer. Math. Soc., 70 (1951), 387–403.Google Scholar
20. Rudin, W., A theorem on subharmonic functions, Proc Amer. Math. Soc., 2 (1951), 209–212.Google Scholar
21. Rudin, W., Inversion of second order differential operators, Proc. Amer. Math. Soc., 8 (1952), 92–98.Google Scholar
22. Saks, S., On the operators of Blaschke and Privaloff for subharmonic functions, Mat. Sb., 9 (1941), 451–456.Google Scholar
23. Zygmund, A., Trigonometrical series, Vol. 1 (Cambridge, 1959).Google Scholar