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A theorem on absolute summability of Fourier series by Riesz means

Published online by Cambridge University Press:  17 April 2009

Prem Chandra  and
S. Izumi
Prem Chandra
Affiliation:
Government Science College, Jabalpur, India.
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Abstract

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In 1951 Mohanty established the following theorem.

If Φ(t)log is of bounded variation in (0, π), wherek ≥ πe2and, then is summable, for however large positive Δ. In this present note we have generalised the above theorem by taking a more general type of Riesz means and under the condition, is of bounded variation in (0, π), where c is finite, imposed upon the generating function of Fourier series.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 3 , Issue 2 , October 1970 , pp. 179 - 184
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Mohanty, R., “On the absolute Riesz summability of Fourier series and allied series”, Proc. London Math. Soc. (2) 52 (1951), 295–320.Google Scholar
[2]Obrechkoff, Nicolas, “Sur la sommation absolue des séries de Dirichlet”, C.R. Acad. Sci. Paris 186 (1928), 215–217.Google Scholar
[3]Obreschkoff, Nikola, “Über die absolute Summierung der Dirichletschen Reihen”, Math. Z. 30 (1929), 375–385.CrossRefGoogle Scholar