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On the Generality of the AP-Integral

Published online by Cambridge University Press:  20 November 2018

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

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 3 , June 1971 , pp. 557 - 561
Copyright
Copyright © Canadian Mathematical Society 1971

References

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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. 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