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On the non-absolute summability of a Fourier series and the conjugate of a Fourier series by a Nörlund method

Published online by Cambridge University Press:  24 October 2008

R. Mohanty  and
B. K. Ray
R. Mohanty
Affiliation:
Ravenshaw College, Cuttack, India
B. K. Ray
Affiliation:
Ravenshaw College, Cuttack, India
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Extract

Let {Sn} be the sequence of partial sums of the infinite seriesΣαn. Let {pn} be a sequence of constants real or complex and let us set

The sequence {tn} of Nörlund means (5) or simply (N, pn) means of the sequence {Sn} generated by the sequence of coefficients {pn} is defined by the following sequence -to-sequence transformation

The series ∑αn or the sequence {Sn} is said to be summable (N, pn) to the sum S, if

and is said to be absolutely summable (N, pn) or summable |N, pn|, if the sequence {tn} is of bounded variation, that is, the series ∑|tn − tn−1| is convergent (2).

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 63 , Issue 2 , April 1967 , pp. 407 - 411
Copyright
Copyright © Cambridge Philosophical Society 1967

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References

REFERENCES

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