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On a convergence test of Hardy-Littlewood's type for fourier series

Published online by Cambridge University Press:  09 April 2009

Fu Cheng Hsiang
Fu Cheng Hsiang
Affiliation:
National Taiwan University, Formosa, China.
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Let ф(t) be an even function integrable in the Lebesgue sense and periodic with period 2п. Let Write By an indirect method based on the method of Riesz summability for the Fourier series, the author has established the following convergence test for the oscifiating series Σan. The theorem is as follows:

THEOREM A [1]. If, for some Δ > 0 as t → +0 andthen фan converges to the sums = 0. Here K is an absolute constant independent of n.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 4 , Issue 1 , February 1964 , pp. 49 - 55
Copyright
Copyright © Australian Mathematical Society 1964

References

[1]Hsiang, F. C., On Riesz summability of Fourier series, Proc. American Math. Soc., 9 (1958), 37–44.Google Scholar
[2]Zygmund, A., Trigonometric series (Cambridge, 1959).Google Scholar