Geodesic
On the Generality of the AP-Integral
Canadian journal of mathematics, Tome 23 (1971) no. 3, pp. 557-561

Voir la notice de l'article provenant de la source Cambridge University Press

In 1955 Taylor [6] constructed an AP-integral sufficiently strong to integrate Abel summable series with coefficients o(n). He showed that the AP-integral includes the special Denjoy integral and further that, when applied to trigonometric series, the AP-integral is more powerful than the SCP-integral of Burkill [1] and the P2-integral of James [3]. The present paper shows that the AP-integral includes the SCP-integral, and, under natural assumptions, the P2-integral.After completing this manuscript I was advised by Skvorcov that he had shown [5] under more general conditions that the P2-integral is included in the AP-integral. The proof in the present paper seems to have some value in its own right and is considerably shorter.Since the definition of the AP-integral is essentially for a function defined in (0, 2π] and elsewhere by 2π-periodicity, we shall consider SCP-integrable and P2-integrable functions defined similarly. 
Cross, G. E. On the Generality of the AP-Integral. Canadian journal of mathematics, Tome 23 (1971) no. 3, pp. 557-561. doi: 10.4153/CJM-1971-062-8
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[1] 1. Burkill, J. C., Integrals and trigonometric series, Proc. London Math. Soc. (3) 1 (1951), 46–57. Google Scholar

[2] 2. Cross, G. E., The relation between two symmetric integrals, Proc. Amer. Math. Soc. H (1963), 185–190. Google Scholar

[3] 3. James, R. D., A generalized integral. II, Can. J. Math. 2 (1950), 297–306. Google Scholar

[4] 4. Skvorcov, V. A., Concerning definitions of P2-and SOP-integrals, Vestnik Moscov. Univ. Ser. I Mat. Meh. 21 no. 6 (1966), 12–19. (Russian) Google Scholar

[5] 5. Skvorcov, V. A., The mutual relationship between the AP-integral of Taylor and the P2-integral of James, Mat. Sb. 170 (112), no. 3, (1966), 380–393. (Russian) Google Scholar

[6] 6. Taylor, S. J., An integral of Perron's type defined with the help of trigonometric series, Quart. J. Math. Oxford Ser. (2) 6 (1955), 255–274. Google Scholar

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