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Convergence criteria for Fourier series

Published online by Cambridge University Press:  17 April 2009

V. Venu Gopal Rao
V. Venu Gopal Rao
Affiliation:
Institute of Advanced Studies, Australian National University, Canberra, ACT.
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Abstract

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The following convergence criterion of Fourier series is due to M. Izumi, S. Izumi and the author:

THEOREM. Let Δ ≥ 1. If

(i) , and

(ii) as t → 0

for an a, 0 < a < 1 and for a δ, 0 < δ < π, then the Fourier series of φ(t) is convergent at the origin.

The object of this paper is to generalize the above theorem in the Hardy-Littlewood direction.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 4 , Issue 2 , April 1971 , pp. 259 - 262
Copyright
Copyright © Australian Mathematical Society 1971

References

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[2]Izumi, Masako, Izumi, Shin-ichi and Gopal Rao, V.V., “On the convergence criteria of Fourier series”, Proc. Japan Acad. (to appear).Google Scholar
[3]Pollard, S., “On the criteria for the convergence of a Fourier series”, J. London Math. Soc. 2 (1927), 255–262.CrossRefGoogle Scholar
[4]Sunouchi, Gen-ichirô, “Notes on Fourier analysis. XLVI. A convergence criterion for Fourier series”, Tôhoku Math. J. (2) 3 (1951), 216–219.Google Scholar